Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12L1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.36.1.72 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&5\\2&21\end{bmatrix}$, $\begin{bmatrix}3&11\\14&3\end{bmatrix}$, $\begin{bmatrix}7&0\\12&11\end{bmatrix}$, $\begin{bmatrix}9&23\\22&21\end{bmatrix}$, $\begin{bmatrix}11&23\\2&1\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $16$ |
| Cyclic 24-torsion field degree: | $128$ |
| Full 24-torsion field degree: | $2048$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x^{2} - z w $ |
| $=$ | $6 y^{2} - 4 z^{2} + 2 z w - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - x^{2} z^{2} - 6 y^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\,\frac{(2z^{3}+w^{3})^{3}}{w^{3}z^{6}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.18.0.k.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.0.a.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.18.1.j.1 | $24$ | $2$ | $2$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 24.72.3.e.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.el.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.eo.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.eu.1 | $24$ | $2$ | $2$ | $3$ | $3$ | $1^{2}$ |
| 24.72.3.rc.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.re.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.rq.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.rs.1 | $24$ | $2$ | $2$ | $3$ | $3$ | $1^{2}$ |
| 24.72.3.bbj.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 24.72.3.bbl.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.bfb.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.bfd.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 24.72.5.dd.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.df.1 | $24$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
| 24.72.5.iv.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 24.72.5.ix.1 | $24$ | $2$ | $2$ | $5$ | $2$ | $1^{4}$ |
| 72.108.5.bj.1 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.324.21.ba.1 | $72$ | $9$ | $9$ | $21$ | $?$ | not computed |
| 120.72.3.eqe.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eqg.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eqs.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.equ.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.esi.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.esk.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.esw.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.esy.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.gqz.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.grb.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.gsf.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.gsh.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.5.bhh.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bhj.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bjd.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bjf.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.180.13.brm.1 | $120$ | $5$ | $5$ | $13$ | $?$ | not computed |
| 120.216.13.bxg.1 | $120$ | $6$ | $6$ | $13$ | $?$ | not computed |
| 168.72.3.eem.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.eeo.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.efa.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.efc.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.egq.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.egs.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ehe.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ehg.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ftv.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ftx.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fvb.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fvd.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.5.sb.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.sd.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.tx.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.tz.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.288.21.bcg.1 | $168$ | $8$ | $8$ | $21$ | $?$ | not computed |
| 264.72.3.eem.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.eeo.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.efa.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.efc.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.egq.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.egs.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ehe.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ehg.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ftv.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ftx.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fvb.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fvd.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.5.sb.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.sd.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.tx.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.tz.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.3.eem.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.eeo.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.efa.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.efc.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.egq.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.egs.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ehe.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ehg.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ftv.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ftx.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fvb.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fvd.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.5.sb.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.sd.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.tx.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.tz.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |