Invariants
| Level: | $60$ | $\SL_2$-level: | $20$ | Newform level: | $3600$ | ||
| Index: | $144$ | $\PSL_2$-index: | $72$ | ||||
| Genus: | $1 = 1 + \frac{ 72 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $2$ are rational) | Cusp widths | $1^{4}\cdot4^{2}\cdot5^{4}\cdot20^{2}$ | Cusp orbits | $1^{2}\cdot2^{3}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 20H1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.144.1.25 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}3&5\\50&11\end{bmatrix}$, $\begin{bmatrix}13&55\\0&31\end{bmatrix}$, $\begin{bmatrix}31&35\\26&21\end{bmatrix}$, $\begin{bmatrix}51&5\\8&21\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 60.72.1.ch.2 for the level structure with $-I$) |
| Cyclic 60-isogeny field degree: | $8$ |
| Cyclic 60-torsion field degree: | $32$ |
| Full 60-torsion field degree: | $15360$ |
Jacobian
| Conductor: | $2^{4}\cdot3^{2}\cdot5^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 3600.2.a.be |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 300x + 1375 $ |
Rational points
This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.
Maps to other modular curves
$j$-invariant map of degree 72 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{3^{10}\cdot5^{10}}\cdot\frac{360x^{2}y^{22}-359235000x^{2}y^{20}z^{2}+25047984375000x^{2}y^{18}z^{4}-535634401640625000x^{2}y^{16}z^{6}+5578106457128906250000x^{2}y^{14}z^{8}-35921626899323730468750000x^{2}y^{12}z^{10}+162803505422546081542968750000x^{2}y^{10}z^{12}-538027203978233528137207031250000x^{2}y^{8}z^{14}+1311119441471756165027618408203125000x^{2}y^{6}z^{16}-2328113527166189990937709808349609375000x^{2}y^{4}z^{18}+2743198323428009059242904186248779296875000x^{2}y^{2}z^{20}-1937829493320453428928740322589874267578125000x^{2}z^{22}-57600xy^{22}z+18391050000xy^{20}z^{3}-848761031250000xy^{18}z^{5}+14924357957812500000xy^{16}z^{7}-140122931009765625000000xy^{14}z^{9}+850238949306225585937500000xy^{12}z^{11}-3689077059225741577148437500000xy^{10}z^{13}+11768615608828697204589843750000000xy^{8}z^{15}-27865250039580153751373291015625000000xy^{6}z^{17}+48027759972020988598465919494628906250000xy^{4}z^{19}-55244323869654530059173703193664550781250000xy^{2}z^{21}+37343192808803479975741356611251831054687500000xz^{23}-y^{24}+5449500y^{22}z^{2}-690800906250y^{20}z^{4}+19566217617187500y^{18}z^{6}-244849893325927734375y^{16}z^{8}+1807439534004638671875000y^{14}z^{10}-9219976939629684448242187500y^{12}z^{12}+34352974558941387176513671875000y^{10}z^{14}-95245863678176437318325042724609375y^{8}z^{16}+197314977408298259943723678588867187500y^{6}z^{18}-290933783284935426140204071998596191406250y^{4}z^{20}+287493417738688892004545778036117553710937500y^{2}z^{22}-138272410875415138725787983275949954986572265625z^{24}}{z^{6}y^{4}(y^{2}-3375z^{2})^{2}(x^{2}y^{8}-499500x^{2}y^{6}z^{2}+12324656250x^{2}y^{4}z^{4}-80115960937500x^{2}y^{2}z^{6}+151154483642578125x^{2}z^{8}-160xy^{8}z+18663750xy^{6}z^{3}-319734843750xy^{4}z^{5}+1729566738281250xy^{2}z^{7}-2912805285644531250xz^{9}+10900y^{8}z^{2}-465075000y^{6}z^{4}+4276325390625y^{4}z^{6}-12872758886718750y^{2}z^{8}+10785164337158203125z^{10})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_1(10)$ | $10$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 60.72.0-10.a.2.12 | $60$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 60.288.5-60.bk.1.4 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.288.5-60.cq.1.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.288.5-60.kj.1.2 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.288.5-60.kl.1.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.288.5-60.oo.1.2 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.288.5-60.os.1.2 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
| 60.288.5-60.qa.1.2 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.288.5-60.qc.1.2 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
| 60.432.13-60.gp.2.3 | $60$ | $3$ | $3$ | $13$ | $1$ | $1^{6}\cdot2^{3}$ |
| 60.576.13-60.nt.2.2 | $60$ | $4$ | $4$ | $13$ | $4$ | $1^{6}\cdot2^{3}$ |
| 60.720.13-60.br.1.7 | $60$ | $5$ | $5$ | $13$ | $3$ | $1^{6}\cdot2^{3}$ |
| 120.288.5-120.iq.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.sl.1.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dbp.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.dcd.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.edm.1.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.eeu.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.ens.1.6 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.288.5-120.eog.1.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |