Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $3^{4}\cdot12^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ 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: | 12K1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.7 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&4\\4&9\end{bmatrix}$, $\begin{bmatrix}7&12\\0&23\end{bmatrix}$, $\begin{bmatrix}7&18\\6&13\end{bmatrix}$, $\begin{bmatrix}21&4\\8&21\end{bmatrix}$, $\begin{bmatrix}23&3\\0&7\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $64$ |
| Full 24-torsion field degree: | $2048$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.f |
Models
Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{3} - 60x - 176 $ |
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 |
|---|
| $(-4:0:1)$, $(0:1:0)$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^6}\cdot\frac{36x^{2}y^{10}+201312x^{2}y^{8}z^{2}+271309824x^{2}y^{6}z^{4}+147787794432x^{2}y^{4}z^{6}+35688827142144x^{2}y^{2}z^{8}+3188955665203200x^{2}z^{10}+828xy^{10}z+2827008xy^{8}z^{3}+3102706944xy^{6}z^{5}+1503847047168xy^{4}z^{7}+336439189979136xy^{2}z^{9}+28471638743580672xz^{11}+y^{12}+13392y^{10}z^{2}+25418880y^{8}z^{4}+18444063744y^{6}z^{6}+6304806678528y^{4}z^{8}+1021402934083584y^{2}z^{10}+62863259171291136z^{12}}{z^{4}(36x^{2}y^{6}+154368x^{2}y^{4}z^{2}+128618496x^{2}y^{2}z^{4}+28835315712x^{2}z^{6}+780xy^{6}z+1961856xy^{4}z^{3}+1309298688xy^{2}z^{5}+257447559168xz^{7}+y^{8}+11408y^{6}z^{2}+14363136y^{4}z^{4}+5409718272y^{2}z^{6}+568425185280z^{8})}$ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(3)$ | $3$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
| 8.6.0.d.1 | $8$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 6.18.0.b.1 | $6$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.0.e.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.1.d.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.72.1.t.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.u.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.w.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.1.x.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.72.3.bf.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.du.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.ht.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.hy.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.mt.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.mu.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.na.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.nb.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.qh.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.qi.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.qk.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.ql.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 72.108.4.d.1 | $72$ | $3$ | $3$ | $4$ | $?$ | not computed |
| 72.108.7.bz.1 | $72$ | $3$ | $3$ | $7$ | $?$ | not computed |
| 120.72.1.bf.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.bg.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.bi.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.1.bj.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.72.3.bml.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bmn.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bmz.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bnb.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bop.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bor.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bpd.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.bpf.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.cgx.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.cgy.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.cha.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.chb.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.180.13.cr.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.216.13.fj.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
| 168.72.1.t.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.u.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.w.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.1.x.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.72.3.bkd.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bkf.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bkr.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bkt.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bmh.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bmj.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bmv.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.bmx.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.cdr.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.cds.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.cdu.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.cdv.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.288.19.ti.1 | $168$ | $8$ | $8$ | $19$ | $?$ | not computed |
| 264.72.1.t.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.u.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.w.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.1.x.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.72.3.bkd.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bkf.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bkr.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bkt.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bmh.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bmj.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bmv.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.bmx.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.cdr.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.cds.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.cdu.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.cdv.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.1.t.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.u.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.w.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.1.x.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.72.3.bkd.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bkf.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bkr.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bkt.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bmh.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bmj.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bmv.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.bmx.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.cdr.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.cds.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.cdu.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.cdv.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |