Invariants
| Level: | $60$ | $\SL_2$-level: | $30$ | Newform level: | $80$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 2 }{2}$ | ||||||
| Cusps: | $2$ (all of which are rational) | Cusp widths | $6\cdot30$ | Cusp orbits | $1^{2}$ | ||
| Elliptic points: | $8$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 30C1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 60.36.1.47 |
Level structure
| $\GL_2(\Z/60\Z)$-generators: | $\begin{bmatrix}21&40\\43&27\end{bmatrix}$, $\begin{bmatrix}38&25\\7&1\end{bmatrix}$, $\begin{bmatrix}46&25\\35&52\end{bmatrix}$, $\begin{bmatrix}53&20\\59&49\end{bmatrix}$, $\begin{bmatrix}58&35\\17&17\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 60-isogeny field degree: | $24$ |
| Cyclic 60-torsion field degree: | $384$ |
| Full 60-torsion field degree: | $61440$ |
Jacobian
| Conductor: | $2^{4}\cdot5$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 80.2.a.b |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - x^{2} + 4x - 4 $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Weierstrass model |
|---|
| $(1:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{12x^{2}y^{10}+305x^{2}y^{8}z^{2}+2540x^{2}y^{6}z^{4}+2151x^{2}y^{4}z^{6}-7772x^{2}y^{2}z^{8}-3481x^{2}z^{10}-54xy^{10}z-868xy^{8}z^{3}-1375xy^{6}z^{5}+13674xy^{4}z^{7}+19435xy^{2}z^{9}-1728xz^{11}-y^{12}+52y^{10}z^{2}-28y^{8}z^{4}-4395y^{6}z^{6}-16603y^{4}z^{8}-9925y^{2}z^{10}+5084z^{12}}{z^{10}(2x^{2}+xz-y^{2}-3z^{2})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_0(5)$ | $5$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 12.6.0.h.1 | $12$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.6.0.h.1 | $12$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 15.18.0.a.1 | $15$ | $2$ | $2$ | $0$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 60.72.1.fp.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.1.fp.2 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.1.fr.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.1.fr.2 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.1.fs.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.1.fs.2 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.1.fu.1 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.1.fu.2 | $60$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 60.72.5.j.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 60.72.5.k.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 60.72.5.y.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 60.72.5.z.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 60.72.5.di.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 60.72.5.dj.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 60.72.5.dl.1 | $60$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 60.72.5.dm.1 | $60$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 60.72.5.eh.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 60.72.5.eh.2 | $60$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 60.72.5.ei.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 60.72.5.ei.2 | $60$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 60.72.5.ek.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 60.72.5.ek.2 | $60$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 60.72.5.el.1 | $60$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 60.72.5.el.2 | $60$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 60.108.5.m.1 | $60$ | $3$ | $3$ | $5$ | $0$ | $1^{4}$ |
| 60.144.7.uk.1 | $60$ | $4$ | $4$ | $7$ | $1$ | $1^{6}$ |
| 60.180.11.bo.1 | $60$ | $5$ | $5$ | $11$ | $4$ | $1^{8}\cdot2$ |
| 120.72.1.sp.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.sp.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.sv.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.sv.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.tb.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.tb.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.th.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.th.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.5.kf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.ki.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.mn.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.mq.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bes.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bey.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bfe.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bfk.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cka.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cka.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.ckg.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.ckg.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.ckm.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.ckm.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cks.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.cks.2 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 180.108.7.fa.1 | $180$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 180.324.19.fi.1 | $180$ | $9$ | $9$ | $19$ | $?$ | not computed |
| 300.180.11.c.1 | $300$ | $5$ | $5$ | $11$ | $?$ | not computed |