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: | $20$ (of which $4$ are rational) | Cusp widths | $4^{8}\cdot8^{2}\cdot12^{8}\cdot24^{2}$ | Cusp orbits | $1^{4}\cdot2^{4}\cdot4^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24AG7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.384.7.1447 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&2\\12&13\end{bmatrix}$, $\begin{bmatrix}5&16\\0&23\end{bmatrix}$, $\begin{bmatrix}7&22\\0&5\end{bmatrix}$, $\begin{bmatrix}23&8\\0&1\end{bmatrix}$, $\begin{bmatrix}23&10\\0&13\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_{12}:C_2^4$ |
| Contains $-I$: | no $\quad$ (see 24.192.7.bc.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $16$ |
| 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, 24.2.d.a, 288.2.d.b, 576.2.a.b, 576.2.a.d |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x v + y t - y u $ |
| $=$ | $y t + y u - z w$ | |
| $=$ | $x t + 2 y z - w u$ | |
| $=$ | $x u - 2 y v - w t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 6 x^{6} z^{4} + 9 x^{4} y^{4} z^{2} - 18 x^{4} y^{2} z^{4} + x^{4} z^{6} - 54 x^{2} y^{6} z^{2} + \cdots + y^{4} z^{6} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:1:0:1:-1:1)$, $(0:0:1:0:-1:1:1)$, $(0:0:-1:0:-1:-1:1)$, $(0:0:-1:0:1:1:1)$ |
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.bo.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -2y+z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -2y+z+v$ |
Equation of the image curve:
| $0$ | $=$ | $ 6X^{4}-4X^{3}Y+6X^{2}Y^{2}+4XY^{3}-8X^{3}Z-6X^{2}YZ+2Y^{3}Z-3X^{2}Z^{2}-6XYZ^{2}-3Y^{2}Z^{2}+2XZ^{3}+YZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.7.bc.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x-w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2z$ |
Equation of the image curve:
| $0$ | $=$ | $ 6X^{6}Z^{4}+9X^{4}Y^{4}Z^{2}-18X^{4}Y^{2}Z^{4}+X^{4}Z^{6}-54X^{2}Y^{6}Z^{2}+24X^{2}Y^{4}Z^{4}-2X^{2}Y^{2}Z^{6}-54Y^{10}+45Y^{8}Z^{2}-12Y^{6}Z^{4}+Y^{4}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.m.2.4 | $24$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 24.192.3-24.bf.1.5 | $24$ | $2$ | $2$ | $3$ | $1$ | $2^{2}$ |
| 24.192.3-24.bf.1.14 | $24$ | $2$ | $2$ | $3$ | $1$ | $2^{2}$ |
| 24.192.3-24.bo.2.8 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 24.192.3-24.bo.2.55 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 24.192.3-24.bq.2.38 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 24.192.3-24.bq.2.61 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
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.bp.1.7 | $24$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 24.768.13-24.bp.2.5 | $24$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 24.768.13-24.br.2.11 | $24$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 24.768.13-24.br.3.9 | $24$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 24.768.13-24.cf.1.5 | $24$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 24.768.13-24.cf.4.1 | $24$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 24.768.13-24.ch.2.5 | $24$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 24.768.13-24.ch.4.1 | $24$ | $2$ | $2$ | $13$ | $1$ | $2^{3}$ |
| 24.768.17-24.dy.2.8 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 24.768.17-24.ga.2.8 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 24.768.17-24.hn.2.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 24.768.17-24.hr.2.4 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 24.768.17-24.lm.1.6 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{3}\cdot4$ |
| 24.768.17-24.lm.2.6 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{3}\cdot4$ |
| 24.768.17-24.lm.3.6 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{3}\cdot4$ |
| 24.768.17-24.lm.4.6 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{3}\cdot4$ |
| 24.768.17-24.ls.1.8 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{3}\cdot4$ |
| 24.768.17-24.ls.2.8 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{3}\cdot4$ |
| 24.768.17-24.ls.3.8 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{3}\cdot4$ |
| 24.768.17-24.ls.4.8 | $24$ | $2$ | $2$ | $17$ | $1$ | $2^{3}\cdot4$ |
| 24.768.17-24.np.2.8 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 24.768.17-24.nx.2.8 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 24.768.17-24.og.2.4 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{6}\cdot2^{2}$ |
| 24.768.17-24.oi.2.4 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{6}\cdot2^{2}$ |
| 24.1152.29-24.fd.1.9 | $24$ | $3$ | $3$ | $29$ | $3$ | $1^{10}\cdot2^{6}$ |