Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $576$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $7 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
| Cusps: | $20$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{2}\cdot6^{4}\cdot8^{4}\cdot12^{2}\cdot24^{4}$ | Cusp orbits | $2^{6}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24AI7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.384.7.1781 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&21\\0&11\end{bmatrix}$, $\begin{bmatrix}5&10\\0&23\end{bmatrix}$, $\begin{bmatrix}13&5\\0&5\end{bmatrix}$, $\begin{bmatrix}19&23\\0&7\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $D_8:D_6$ |
| Contains $-I$: | no $\quad$ (see 24.192.7.el.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $1$ |
| Cyclic 24-torsion field degree: | $8$ |
| Full 24-torsion field degree: | $192$ |
Jacobian
| Conductor: | $2^{31}\cdot3^{11}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2^{2}$ |
| Newforms: | 24.2.a.a, 72.2.d.b, 96.2.d.a, 576.2.a.b, 576.2.a.d |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x t - x v + y t + y u - w t - w v $ |
| $=$ | $x t + x u - y t + y v - w t + w u$ | |
| $=$ | $x z + y u - y v - z w - w u - w v$ | |
| $=$ | $2 x t + x u - x v - y z$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} y^{2} + 3 x^{6} z^{2} + 2 x^{4} y^{4} + 18 x^{4} y^{2} z^{2} + 36 x^{4} z^{4} + \cdots + 216 y^{2} z^{6} $ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=47$, and therefore no rational points.
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 24.96.3.gf.2 :
| $\displaystyle X$ | $=$ | $\displaystyle -x+w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x+w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -y$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{3}Y-XY^{3}-X^{2}Z^{2}-Y^{2}Z^{2}-2Z^{4} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.7.el.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{6}Y^{2}+3X^{6}Z^{2}+2X^{4}Y^{4}+18X^{4}Y^{2}Z^{2}+36X^{4}Z^{4}-24X^{2}Y^{4}Z^{2}+108X^{2}Y^{2}Z^{4}+108X^{2}Z^{6}+72Y^{4}Z^{4}+216Y^{2}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.96.0-24.bk.1.3 | $24$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 24.192.3-24.fb.1.2 | $24$ | $2$ | $2$ | $3$ | $1$ | $2^{2}$ |
| 24.192.3-24.fb.1.18 | $24$ | $2$ | $2$ | $3$ | $1$ | $2^{2}$ |
| 24.192.3-24.gf.2.10 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 24.192.3-24.gf.2.13 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 24.192.3-24.gh.2.12 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 24.192.3-24.gh.2.25 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.768.13-24.fs.2.2 | $24$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 24.768.13-24.fs.4.4 | $24$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 24.768.13-24.fu.3.1 | $24$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 24.768.13-24.fu.4.2 | $24$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 24.768.13-24.fw.1.3 | $24$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 24.768.13-24.fw.3.7 | $24$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 24.768.13-24.fy.1.1 | $24$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 24.768.13-24.fy.2.3 | $24$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 24.1152.29-24.ld.1.10 | $24$ | $3$ | $3$ | $29$ | $3$ | $1^{10}\cdot2^{6}$ |
| 48.768.17-48.we.2.6 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.wi.2.6 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.wu.1.6 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.wy.1.6 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.bej.4.9 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{3}\cdot4$ |
| 48.768.17-48.bej.5.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{3}\cdot4$ |
| 48.768.17-48.bej.7.9 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{3}\cdot4$ |
| 48.768.17-48.bej.8.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{3}\cdot4$ |
| 48.768.17-48.bep.1.3 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{3}\cdot4$ |
| 48.768.17-48.bep.2.7 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{3}\cdot4$ |
| 48.768.17-48.bep.3.5 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{3}\cdot4$ |
| 48.768.17-48.bep.6.13 | $48$ | $2$ | $2$ | $17$ | $1$ | $2^{3}\cdot4$ |
| 48.768.17-48.bky.1.3 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.blc.1.3 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.blo.2.5 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.bls.2.5 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |