Modular polynomials


This page lists the modular polynomials ΦN(X,Y) for levels N up to to 300; 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 these polynomials the coefficient of XmYn is always the same as that of XnYm so only the coefficients of terms with m ≥ n are listed).

For example, the file for Φ4(X,Y) looks like

[3,0] 1
[2,0] −16200
[2,1] 1488
[2,2] −1
[1,0] 8748000000
[1,1] 40773375
[0,0] −157464000000000

which indicates the polynomial

X3X2Y2 + 1488X2Y − 162000X2 + 14888XY2+ 4077375XY + 8748000000X + Y3 − 162000Y2 + 8748000000Y −157464000000000.

Invididual links each polynomial are provided below.
A single compressed tar file containing all these files 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)