Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $6^{8}\cdot24^{4}$ | Cusp orbits | $2^{6}$ | ||
| 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: | 24J7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.288.7.1221 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&0\\0&13\end{bmatrix}$, $\begin{bmatrix}7&14\\20&5\end{bmatrix}$, $\begin{bmatrix}9&2\\16&9\end{bmatrix}$, $\begin{bmatrix}13&18\\0&1\end{bmatrix}$, $\begin{bmatrix}23&22\\20&5\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4^2.C_2^4$ |
| Contains $-I$: | no $\quad$ (see 24.144.7.il.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $16$ |
| Full 24-torsion field degree: | $256$ |
Jacobian
| Conductor: | $2^{21}\cdot3^{11}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}$ |
| Newforms: | 24.2.a.a, 36.2.a.a$^{3}$, 48.2.a.a$^{2}$, 144.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x^{2} + z^{2} + w t $ |
| $=$ | $w^{2} + t^{2} + u^{2} + v^{2}$ | |
| $=$ | $2 x z + w v - t u$ | |
| $=$ | $2 x^{2} - w t - u v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{10} + 2 x^{6} z^{4} + 12 x^{4} y^{2} z^{4} + 12 x^{2} y^{4} z^{4} + x^{2} z^{8} + 4 y^{6} z^{4} $ |
Rational points
This modular curve has no real points and no $\Q_p$ points for $p=19$, 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.q.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -v$ |
| $\displaystyle W$ | $=$ | $\displaystyle -u$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{2}-4Y^{2}-ZW $ |
| $=$ | $ 3X^{3}-YZ^{2}-2XZW+YW^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.144.7.il.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{10}+2X^{6}Z^{4}+12X^{4}Y^{2}Z^{4}+12X^{2}Y^{4}Z^{4}+X^{2}Z^{8}+4Y^{6}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.144.3-12.l.1.2 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.3-12.l.1.15 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.3-24.ov.1.3 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.3-24.ov.1.22 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.3-24.pw.1.3 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.3-24.pw.1.14 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.4-24.q.1.22 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.q.1.23 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.ch.1.47 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.gp.1.3 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.gp.1.21 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.hi.1.4 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.hi.1.14 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
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.om.1.3 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.om.2.1 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.on.1.7 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.on.2.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.oo.1.7 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.oo.2.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.op.1.4 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.op.2.3 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.oq.1.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.oq.2.1 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.or.1.4 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.or.2.3 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.17-24.uv.1.23 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
| 24.576.17-24.wc.1.15 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
| 24.576.17-24.bnh.1.13 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
| 24.576.17-24.bni.1.11 | $24$ | $2$ | $2$ | $17$ | $4$ | $1^{10}$ |
| 48.576.15-48.cc.1.7 | $48$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 48.576.15-48.cc.2.13 | $48$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 48.576.15-48.cu.1.3 | $48$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 48.576.15-48.cu.2.7 | $48$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 48.576.17-48.cz.1.7 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
| 48.576.17-48.cz.2.6 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
| 48.576.17-48.db.1.7 | $48$ | $2$ | $2$ | $17$ | $4$ | $1^{10}$ |
| 48.576.17-48.db.2.6 | $48$ | $2$ | $2$ | $17$ | $4$ | $1^{10}$ |
| 48.576.17-48.dc.1.13 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
| 48.576.17-48.dc.2.11 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
| 48.576.17-48.dd.1.3 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
| 48.576.17-48.dd.2.3 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
| 48.576.19-48.kq.1.7 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{2}\cdot4^{2}$ |
| 48.576.19-48.kq.2.13 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{2}\cdot4^{2}$ |
| 48.576.19-48.nu.1.3 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{2}\cdot4^{2}$ |
| 48.576.19-48.nu.2.7 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{2}\cdot4^{2}$ |