Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $8 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (none of which are rational) | Cusp widths | $6^{4}\cdot12^{2}\cdot24^{4}$ | Cusp orbits | $2^{5}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 6$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24D8 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.288.8.2107 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&16\\16&13\end{bmatrix}$, $\begin{bmatrix}11&10\\16&13\end{bmatrix}$, $\begin{bmatrix}11&12\\0&1\end{bmatrix}$, $\begin{bmatrix}17&4\\8&17\end{bmatrix}$, $\begin{bmatrix}17&6\\0&17\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4.D_4^2$ |
| Contains $-I$: | no $\quad$ (see 24.144.8.fg.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $256$ |
Jacobian
| Conductor: | $2^{22}\cdot3^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}\cdot2^{2}$ |
| Newforms: | 36.2.a.a$^{3}$, 72.2.d.b$^{2}$, 144.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x^{2} + x y - x z + x w - x v - y z - y t + z^{2} - z t + z u - z v - w u - t u + t v $ |
| $=$ | $x y - x z + x w + 2 x u - x r - y z + y t - 2 y u + z^{2} - z w - z t + z r - t r$ | |
| $=$ | $x^{2} + x y - x z - x w - 2 x u + x v - y z + y t - 2 y u + z w - z v + t v$ | |
| $=$ | $x^{2} - 2 x u - x v + x r + 2 y^{2} - y z - y w - y t + 2 y v + z w - z v - w r + t v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} y^{2} z^{6} + x^{6} z^{8} + 6 x^{5} y^{4} z^{5} - 6 x^{5} z^{9} + 24 x^{4} y^{6} z^{4} + \cdots + 64 z^{14} $ |
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 24.72.4.u.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -x+z+u$ |
| $\displaystyle W$ | $=$ | $\displaystyle x-z+w+u-2r$ |
Equation of the image curve:
| $0$ | $=$ | $ 6X^{2}-18XY+24Y^{2}-ZW+W^{2} $ |
| $=$ | $ 3X^{3}-6XY^{2}+YZ^{2}-XZW-YZW $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.144.8.fg.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x+z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{6}Y^{2}Z^{6}+X^{6}Z^{8}+6X^{5}Y^{4}Z^{5}-6X^{5}Z^{9}+24X^{4}Y^{6}Z^{4}-24X^{4}Y^{4}Z^{6}-24X^{4}Y^{2}Z^{8}+24X^{4}Z^{10}+56X^{3}Y^{8}Z^{3}-58X^{3}Y^{6}Z^{5}+58X^{3}Y^{2}Z^{9}-56X^{3}Z^{11}+96X^{2}Y^{10}Z^{2}-216X^{2}Y^{9}Z^{3}-72X^{2}Y^{8}Z^{4}-24X^{2}Y^{6}Z^{6}+432X^{2}Y^{5}Z^{7}-24X^{2}Y^{4}Z^{8}-72X^{2}Y^{2}Z^{10}-216X^{2}YZ^{11}+96X^{2}Z^{12}+96XY^{12}Z-432XY^{11}Z^{2}+48XY^{10}Z^{3}+432XY^{9}Z^{4}-384XY^{8}Z^{5}+864XY^{7}Z^{6}-864XY^{5}Z^{8}+384XY^{4}Z^{9}-432XY^{3}Z^{10}-48XY^{2}Z^{11}+432XYZ^{12}-96XZ^{13}+64Y^{14}+544Y^{12}Z^{2}+648Y^{11}Z^{3}+36Y^{10}Z^{4}+54Y^{9}Z^{5}-644Y^{8}Z^{6}-1404Y^{7}Z^{7}-644Y^{6}Z^{8}+54Y^{5}Z^{9}+36Y^{4}Z^{10}+648Y^{3}Z^{11}+544Y^{2}Z^{12}+64Z^{14} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.144.4-24.u.1.15 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 24.144.4-24.u.1.31 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
| 24.144.4-24.ch.1.40 | $24$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
| 24.144.4-24.gg.2.7 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 24.144.4-24.gg.2.26 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.576.15-24.kz.1.7 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.la.2.5 | $24$ | $2$ | $2$ | $15$ | $2$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.ls.2.1 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.md.2.5 | $24$ | $2$ | $2$ | $15$ | $1$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.nr.2.15 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.nv.1.11 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.on.1.7 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.ow.1.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.17-24.lj.1.22 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.ti.1.12 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.bla.1.12 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.bli.1.12 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.brq.2.5 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.bry.1.7 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.bti.2.5 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.btq.1.8 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.k.1.14 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.bf.2.13 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cd.2.13 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cs.1.13 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dp.1.6 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ds.2.12 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ek.1.6 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.et.1.3 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jf.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.js.1.6 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.mk.1.3 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.mo.1.6 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.ok.1.6 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.pc.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.pm.1.6 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.qn.1.3 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |