accessibility
Modular polynomials
This page lists the modular polynomials ΦN(X,Y) for all levels N up to to 400 and prime levels N up to 1000; these are canonical equations for the modular curve X0(N) that parameterizes pairs of elliptic curves related by a cyclic isogeny of degree N.
For prime levels these polynomials were computed using the algorithm described in Modular polynomials via isogeny volcanoes;
for composite levels N they were computed using the algorithm described in Class polynomials for nonholomorphic modular functions (see Algorithm 1.1 and Corollary 3.4).
If you use these polynomials in your research, please cite one or both of these references, as appropriate (see the list of papers on my home page for full bibliographic details).
Each file is an ascii text file with a line for each non-zero coefficient of ΦN;(X,Y), up to symmetry
(for N>1 the coefficient of XmYn is the same as that of XnYm so for N>1 only the coefficients of terms with m ≥ n are listed).
For example, the file for Φ2(X,Y) looks like
[3,0] 1
[2,0] −162000
[2,1] 1488
[2,2] −1
[1,0] 8748000000
[1,1] 40773375
[0,0] −157464000000000
which indicates the polynomial
X3 − X2Y2 + 1488X2Y
− 162000X2 + 1488XY2+ 40773375XY
+ 8748000000X + Y3 − 162000Y2 + 8748000000Y −157464000000000.
Invididual links for each each polynomial are provided below.
A single compressed tar file containing files for ΦN(X,Y) for all N ≤ 300 may be downloaded from here.
|
Φ1(X,Y) |
Φ2(X,Y) |
Φ3(X,Y) |
Φ4(X,Y) |
Φ5(X,Y) |
Φ6(X,Y) |
Φ7(X,Y) |
Φ8(X,Y) |
Φ9(X,Y) |
Φ10(X,Y) |
|
|
Φ11(X,Y) |
Φ12(X,Y) |
Φ13(X,Y) |
Φ14(X,Y) |
Φ15(X,Y) |
Φ16(X,Y) |
Φ17(X,Y) |
Φ18(X,Y) |
Φ19(X,Y) |
Φ20(X,Y) |
|
|
Φ21(X,Y) |
Φ22(X,Y) |
Φ23(X,Y) |
Φ24(X,Y) |
Φ25(X,Y) |
Φ26(X,Y) |
Φ27(X,Y) |
Φ28(X,Y) |
Φ29(X,Y) |
Φ30(X,Y) |
|
|
Φ31(X,Y) |
Φ32(X,Y) |
Φ33(X,Y) |
Φ34(X,Y) |
Φ35(X,Y) |
Φ36(X,Y) |
Φ37(X,Y) |
Φ38(X,Y) |
Φ39(X,Y) |
Φ40(X,Y) |
|
|
Φ41(X,Y) |
Φ42(X,Y) |
Φ43(X,Y) |
Φ44(X,Y) |
Φ45(X,Y) |
Φ46(X,Y) |
Φ47(X,Y) |
Φ48(X,Y) |
Φ49(X,Y) |
Φ50(X,Y) |
|
|
Φ51(X,Y) |
Φ52(X,Y) |
Φ53(X,Y) |
Φ54(X,Y) |
Φ55(X,Y) |
Φ56(X,Y) |
Φ57(X,Y) |
Φ58(X,Y) |
Φ59(X,Y) |
Φ60(X,Y) |
|
|
Φ61(X,Y) |
Φ62(X,Y) |
Φ63(X,Y) |
Φ64(X,Y) |
Φ65(X,Y) |
Φ66(X,Y) |
Φ67(X,Y) |
Φ68(X,Y) |
Φ69(X,Y) |
Φ70(X,Y) |
|
|
Φ71(X,Y) |
Φ72(X,Y) |
Φ73(X,Y) |
Φ71(X,Y) |
Φ75(X,Y) |
Φ76(X,Y) |
Φ77(X,Y) |
Φ78(X,Y) |
Φ79(X,Y) |
Φ80(X,Y) |
|
|
Φ81(X,Y) |
Φ82(X,Y) |
Φ83(X,Y) |
Φ84(X,Y) |
Φ85(X,Y) |
Φ86(X,Y) |
Φ87(X,Y) |
Φ88(X,Y) |
Φ89(X,Y) |
Φ90(X,Y) |
|
|
Φ91(X,Y) |
Φ92(X,Y) |
Φ93(X,Y) |
Φ94(X,Y) |
Φ95(X,Y) |
Φ96(X,Y) |
Φ97(X,Y) |
Φ98(X,Y) |
Φ99(X,Y) |
Φ100(X,Y) |
|
|
Φ101(X,Y) |
Φ102(X,Y) |
Φ103(X,Y) |
Φ104(X,Y) |
Φ105(X,Y) |
Φ106(X,Y) |
Φ107(X,Y) |
Φ108(X,Y) |
Φ109(X,Y) |
Φ110(X,Y) |
|
|
Φ111(X,Y) |
Φ112(X,Y) |
Φ113(X,Y) |
Φ114(X,Y) |
Φ115(X,Y) |
Φ116(X,Y) |
Φ117(X,Y) |
Φ118(X,Y) |
Φ119(X,Y) |
Φ120(X,Y) |
|
|
Φ121(X,Y) |
Φ122(X,Y) |
Φ123(X,Y) |
Φ124(X,Y) |
Φ125(X,Y) |
Φ126(X,Y) |
Φ127(X,Y) |
Φ128(X,Y) |
Φ129(X,Y) |
Φ130(X,Y) |
|
|
Φ131(X,Y) |
Φ132(X,Y) |
Φ133(X,Y) |
Φ134(X,Y) |
Φ135(X,Y) |
Φ136(X,Y) |
Φ137(X,Y) |
Φ138(X,Y) |
Φ139(X,Y) |
Φ140(X,Y) |
|
|
Φ141(X,Y) |
Φ142(X,Y) |
Φ143(X,Y) |
Φ144(X,Y) |
Φ145(X,Y) |
Φ146(X,Y) |
Φ147(X,Y) |
Φ148(X,Y) |
Φ149(X,Y) |
Φ150(X,Y) |
|
|
Φ151(X,Y) |
Φ152(X,Y) |
Φ153(X,Y) |
Φ154(X,Y) |
Φ155(X,Y) |
Φ156(X,Y) |
Φ157(X,Y) |
Φ158(X,Y) |
Φ159(X,Y) |
Φ160(X,Y) |
|
|
Φ161(X,Y) |
Φ162(X,Y) |
Φ163(X,Y) |
Φ164(X,Y) |
Φ165(X,Y) |
Φ166(X,Y) |
Φ167(X,Y) |
Φ168(X,Y) |
Φ169(X,Y) |
Φ170(X,Y) |
|
|
Φ171(X,Y) |
Φ172(X,Y) |
Φ173(X,Y) |
Φ174(X,Y) |
Φ175(X,Y) |
Φ176(X,Y) |
Φ177(X,Y) |
Φ178(X,Y) |
Φ179(X,Y) |
Φ180(X,Y) |
|
|
Φ181(X,Y) |
Φ182(X,Y) |
Φ183(X,Y) |
Φ184(X,Y) |
Φ185(X,Y) |
Φ186(X,Y) |
Φ187(X,Y) |
Φ188(X,Y) |
Φ189(X,Y) |
Φ190(X,Y) |
|
|
Φ191(X,Y) |
Φ192(X,Y) |
Φ193(X,Y) |
Φ194(X,Y) |
Φ195(X,Y) |
Φ196(X,Y) |
Φ197(X,Y) |
Φ198(X,Y) |
Φ199(X,Y) |
Φ200(X,Y) |
|
|
Φ201(X,Y) |
Φ202(X,Y) |
Φ203(X,Y) |
Φ204(X,Y) |
Φ205(X,Y) |
Φ206(X,Y) |
Φ207(X,Y) |
Φ208(X,Y) |
Φ209(X,Y) |
Φ210(X,Y) |
|
|
Φ211(X,Y) |
Φ212(X,Y) |
Φ213(X,Y) |
Φ214(X,Y) |
Φ215(X,Y) |
Φ216(X,Y) |
Φ217(X,Y) |
Φ218(X,Y) |
Φ219(X,Y) |
Φ220(X,Y) |
|
|
Φ221(X,Y) |
Φ222(X,Y) |
Φ223(X,Y) |
Φ224(X,Y) |
Φ225(X,Y) |
Φ226(X,Y) |
Φ227(X,Y) |
Φ228(X,Y) |
Φ229(X,Y) |
Φ230(X,Y) |
|
|
Φ231(X,Y) |
Φ232(X,Y) |
Φ233(X,Y) |
Φ234(X,Y) |
Φ235(X,Y) |
Φ236(X,Y) |
Φ237(X,Y) |
Φ238(X,Y) |
Φ239(X,Y) |
Φ240(X,Y) |
|
|
Φ241(X,Y) |
Φ242(X,Y) |
Φ243(X,Y) |
Φ244(X,Y) |
Φ245(X,Y) |
Φ246(X,Y) |
Φ247(X,Y) |
Φ248(X,Y) |
Φ249(X,Y) |
Φ250(X,Y) |
|
|
Φ251(X,Y) |
Φ252(X,Y) |
Φ253(X,Y) |
Φ254(X,Y) |
Φ255(X,Y) |
Φ256(X,Y) |
Φ257(X,Y) |
Φ258(X,Y) |
Φ259(X,Y) |
Φ260(X,Y) |
|
|
Φ261(X,Y) |
Φ262(X,Y) |
Φ263(X,Y) |
Φ264(X,Y) |
Φ265(X,Y) |
Φ266(X,Y) |
Φ267(X,Y) |
Φ268(X,Y) |
Φ269(X,Y) |
Φ270(X,Y) |
|
|
Φ271(X,Y) |
Φ272(X,Y) |
Φ273(X,Y) |
Φ274(X,Y) |
Φ275(X,Y) |
Φ276(X,Y) |
Φ277(X,Y) |
Φ278(X,Y) |
Φ279(X,Y) |
Φ280(X,Y) |
|
|
Φ281(X,Y) |
Φ282(X,Y) |
Φ283(X,Y) |
Φ284(X,Y) |
Φ285(X,Y) |
Φ286(X,Y) |
Φ287(X,Y) |
Φ288(X,Y) |
Φ289(X,Y) |
Φ290(X,Y) |
|
|
Φ291(X,Y) |
Φ292(X,Y) |
Φ293(X,Y) |
Φ294(X,Y) |
Φ295(X,Y) |
Φ296(X,Y) |
Φ297(X,Y) |
Φ298(X,Y) |
Φ299(X,Y) |
Φ300(X,Y) |
|
|
Φ301(X,Y) |
Φ302(X,Y) |
Φ303(X,Y) |
Φ304(X,Y) |
Φ305(X,Y) |
Φ306(X,Y) |
Φ307(X,Y) |
Φ308(X,Y) |
Φ309(X,Y) |
Φ310(X,Y) |
|
|
Φ311(X,Y) |
Φ312(X,Y) |
Φ313(X,Y) |
Φ314(X,Y) |
Φ315(X,Y) |
Φ316(X,Y) |
Φ317(X,Y) |
Φ318(X,Y) |
Φ319(X,Y) |
Φ320(X,Y) |
|
|
Φ321(X,Y) |
Φ322(X,Y) |
Φ323(X,Y) |
Φ324(X,Y) |
Φ325(X,Y) |
Φ326(X,Y) |
Φ327(X,Y) |
Φ328(X,Y) |
Φ329(X,Y) |
Φ330(X,Y) |
|
|
Φ331(X,Y) |
Φ332(X,Y) |
Φ333(X,Y) |
Φ334(X,Y) |
Φ335(X,Y) |
Φ336(X,Y) |
Φ337(X,Y) |
Φ338(X,Y) |
Φ339(X,Y) |
Φ340(X,Y) |
|
|
Φ341(X,Y) |
Φ342(X,Y) |
Φ343(X,Y) |
Φ344(X,Y) |
Φ345(X,Y) |
Φ346(X,Y) |
Φ347(X,Y) |
Φ348(X,Y) |
Φ349(X,Y) |
Φ350(X,Y) |
|
|
Φ351(X,Y) |
Φ352(X,Y) |
Φ353(X,Y) |
Φ354(X,Y) |
Φ355(X,Y) |
Φ356(X,Y) |
Φ357(X,Y) |
Φ358(X,Y) |
Φ359(X,Y) |
Φ360(X,Y) |
|
|
Φ361(X,Y) |
Φ362(X,Y) |
Φ363(X,Y) |
Φ364(X,Y) |
Φ365(X,Y) |
Φ366(X,Y) |
Φ367(X,Y) |
Φ368(X,Y) |
Φ369(X,Y) |
Φ370(X,Y) |
|
|
Φ371(X,Y) |
Φ372(X,Y) |
Φ373(X,Y) |
Φ374(X,Y) |
Φ375(X,Y) |
Φ376(X,Y) |
Φ377(X,Y) |
Φ378(X,Y) |
Φ379(X,Y) |
Φ380(X,Y) |
|
|
Φ381(X,Y) |
Φ382(X,Y) |
Φ383(X,Y) |
Φ384(X,Y) |
Φ385(X,Y) |
Φ386(X,Y) |
Φ387(X,Y) |
Φ388(X,Y) |
Φ389(X,Y) |
Φ390(X,Y) |
|
|
Φ391(X,Y) |
Φ392(X,Y) |
Φ393(X,Y) |
Φ394(X,Y) |
Φ395(X,Y) |
Φ396(X,Y) |
Φ397(X,Y) |
Φ398(X,Y) |
Φ399(X,Y) |
Φ400(X,Y) |
|
|
Φ401(X,Y) |
Φ409(X,Y) |
Φ419(X,Y) |
Φ421(X,Y) |
Φ431(X,Y) |
Φ433(X,Y) |
Φ439(X,Y) |
Φ443(X,Y) |
Φ457(X,Y) |
Φ461(X,Y) |
|
|
Φ463(X,Y) |
Φ467(X,Y) |
Φ479(X,Y) |
Φ487(X,Y) |
Φ491(X,Y) |
Φ499(X,Y) |
Φ503(X,Y) |
Φ509(X,Y) |
Φ521(X,Y) |
Φ523(X,Y) |
|
|
Φ541(X,Y) |
Φ547(X,Y) |
Φ557(X,Y) |
Φ563(X,Y) |
Φ569(X,Y) |
Φ571(X,Y) |
Φ577(X,Y) |
Φ587(X,Y) |
Φ593(X,Y) |
Φ599(X,Y) |
|
|
Φ601(X,Y) |
Φ607(X,Y) |
Φ613(X,Y) |
Φ617(X,Y) |
Φ619(X,Y) |
Φ631(X,Y) |
Φ641(X,Y) |
Φ643(X,Y) |
Φ647(X,Y) |
Φ653(X,Y) |
|
|
Φ659(X,Y) |
Φ661(X,Y) |
Φ673(X,Y) |
Φ677(X,Y) |
Φ683(X,Y) |
Φ691(X,Y) |
Φ701(X,Y) |
Φ709(X,Y) |
Φ719(X,Y) |
Φ727(X,Y) |
|
|
Φ733(X,Y) |
Φ739(X,Y) |
Φ743(X,Y) |
Φ751(X,Y) |
Φ757(X,Y) |
Φ761(X,Y) |
Φ769(X,Y) |
Φ773(X,Y) |
Φ787(X,Y) |
Φ797(X,Y) |
|
|
Φ809(X,Y) |
Φ811(X,Y) |
Φ821(X,Y) |
Φ823(X,Y) |
Φ827(X,Y) |
Φ829(X,Y) |
Φ839(X,Y) |
Φ853(X,Y) |
Φ857(X,Y) |
Φ859(X,Y) |
|
|
Φ863(X,Y) |
Φ877(X,Y) |
Φ881(X,Y) |
Φ883(X,Y) |
Φ887(X,Y) |
Φ907(X,Y) |
Φ911(X,Y) |
Φ919(X,Y) |
Φ929(X,Y) |
Φ937(X,Y) |
|
|
Φ941(X,Y) |
Φ947(X,Y) |
Φ953(X,Y) |
Φ967(X,Y) |
Φ971(X,Y) |
Φ977(X,Y) |
Φ983(X,Y) |
Φ991(X,Y) |
Φ997(X,Y) |
|