Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| 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 | $2^{10}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-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: | 24.384.7.823 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&10\\0&5\end{bmatrix}$, $\begin{bmatrix}7&10\\0&1\end{bmatrix}$, $\begin{bmatrix}17&0\\0&17\end{bmatrix}$, $\begin{bmatrix}19&2\\0&1\end{bmatrix}$, $\begin{bmatrix}23&20\\0&17\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_{12}:C_2^4$ |
| Contains $-I$: | no $\quad$ (see 24.192.7.do.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: | $2^{22}\cdot3^{11}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}\cdot2^{2}$ |
| Newforms: | 24.2.a.a$^{2}$, 48.2.a.a, 72.2.d.b$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ w u + t v $ |
| $=$ | $x v - z u$ | |
| $=$ | $x w + z t$ | |
| $=$ | $x u + y v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} y^{2} + 6 x^{6} z^{2} - 2 x^{4} y^{4} + 18 x^{4} y^{2} z^{2} + 36 x^{4} z^{4} + x^{2} y^{6} + \cdots + 36 y^{4} z^{4} $ |
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.bp.2 :
| $\displaystyle X$ | $=$ | $\displaystyle -4x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2x-w-u$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -2x+w-u$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{4}+2X^{3}Y+3X^{2}Y^{2}+XY^{3}-X^{3}Z-3XY^{2}Z-2Y^{3}Z-3XYZ^{2}+XZ^{3}+2YZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.7.do.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{6}Y^{2}+6X^{6}Z^{2}-2X^{4}Y^{4}+18X^{4}Y^{2}Z^{2}+36X^{4}Z^{4}+X^{2}Y^{6}+18X^{2}Y^{4}Z^{2}+72X^{2}Y^{2}Z^{4}+6Y^{6}Z^{2}+36Y^{4}Z^{4} $ |
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.bc.1.3 | $24$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 24.192.3-24.bp.2.2 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 24.192.3-24.bp.2.34 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 24.192.3-24.cl.1.18 | $24$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 24.192.3-24.cl.1.29 | $24$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 24.192.3-24.gh.2.3 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 24.192.3-24.gh.2.30 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
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.dm.1.1 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 24.768.13-24.dm.2.2 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 24.768.13-24.dq.1.2 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 24.768.13-24.dq.2.4 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 24.768.13-24.ek.1.2 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 24.768.13-24.ek.2.4 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 24.768.13-24.eo.2.4 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 24.768.13-24.eo.4.8 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 24.768.17-24.fh.2.11 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2^{2}$ |
| 24.768.17-24.hv.2.1 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 24.768.17-24.oc.2.5 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 24.768.17-24.oh.2.1 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{6}\cdot2^{2}$ |
| 24.768.17-24.qe.3.3 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 24.768.17-24.qe.4.3 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 24.768.17-24.qi.3.5 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 24.768.17-24.qi.4.5 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 24.1152.29-24.ha.2.4 | $24$ | $3$ | $3$ | $29$ | $0$ | $1^{10}\cdot2^{6}$ |
| 48.768.17-48.gt.1.3 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.gt.2.5 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.hb.3.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.hb.4.1 | $48$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 48.768.17-48.hp.2.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.ia.1.2 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.ik.2.6 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{6}\cdot2^{2}$ |
| 48.768.17-48.ip.2.2 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 48.768.21-48.ff.1.4 | $48$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2^{2}\cdot4$ |
| 48.768.21-48.fr.2.8 | $48$ | $2$ | $2$ | $21$ | $1$ | $1^{6}\cdot2^{2}\cdot4$ |
| 48.768.21-48.jh.2.6 | $48$ | $2$ | $2$ | $21$ | $1$ | $1^{6}\cdot2^{2}\cdot4$ |
| 48.768.21-48.kb.2.2 | $48$ | $2$ | $2$ | $21$ | $0$ | $1^{6}\cdot2^{2}\cdot4$ |
| 48.768.21-48.la.3.7 | $48$ | $2$ | $2$ | $21$ | $0$ | $2\cdot4^{3}$ |
| 48.768.21-48.la.4.7 | $48$ | $2$ | $2$ | $21$ | $0$ | $2\cdot4^{3}$ |
| 48.768.21-48.li.3.5 | $48$ | $2$ | $2$ | $21$ | $0$ | $2\cdot4^{3}$ |
| 48.768.21-48.li.4.5 | $48$ | $2$ | $2$ | $21$ | $0$ | $2\cdot4^{3}$ |