Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $288$ | ||
| 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: | 24.288.8.2110 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&10\\8&21\end{bmatrix}$, $\begin{bmatrix}5&4\\16&13\end{bmatrix}$, $\begin{bmatrix}7&6\\0&17\end{bmatrix}$, $\begin{bmatrix}11&8\\8&1\end{bmatrix}$, $\begin{bmatrix}21&20\\16&9\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4.D_4^2$ |
| Contains $-I$: | no $\quad$ (see 24.144.8.fn.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: | $2^{30}\cdot3^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}\cdot2^{2}$ |
| Newforms: | 36.2.a.a$^{3}$, 144.2.a.a, 288.2.d.b$^{2}$ |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x u - x r + 2 y r - w v $ |
| $=$ | $x u - x r + 2 y u - w t$ | |
| $=$ | $x t - x v - 2 y v - z r$ | |
| $=$ | $x t - x v - 2 y t - z u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 2 x^{12} + 4 x^{10} y^{2} + 6 x^{8} y^{4} - 16 x^{6} y^{6} + 36 x^{6} y^{4} z^{2} + \cdots + 27 y^{6} z^{6} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known 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.x.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
| $\displaystyle W$ | $=$ | $\displaystyle v$ |
Equation of the image curve:
| $0$ | $=$ | $ 12X^{2}-36XY+48Y^{2}-ZW-W^{2} $ |
| $=$ | $ 6X^{3}-12XY^{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.fn.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}t$ |
Equation of the image curve:
| $0$ | $=$ | $ -2X^{12}+4X^{10}Y^{2}+6X^{8}Y^{4}-16X^{6}Y^{6}+36X^{6}Y^{4}Z^{2}-36X^{4}Y^{6}Z^{2}+54X^{4}Y^{4}Z^{4}+16X^{2}Y^{10}-36X^{2}Y^{8}Z^{2}+27X^{2}Y^{4}Z^{6}-8Y^{12}+36Y^{10}Z^{2}-54Y^{8}Z^{4}+27Y^{6}Z^{6} $ |
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.x.1.13 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 24.144.4-24.x.1.30 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 24.144.4-24.ch.1.36 | $24$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
| 24.144.4-24.gj.2.7 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 24.144.4-24.gj.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.ku.1.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.lf.1.3 | $24$ | $2$ | $2$ | $15$ | $2$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.lx.2.2 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.ly.1.1 | $24$ | $2$ | $2$ | $15$ | $1$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.nn.1.9 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.ny.1.9 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.oq.1.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.ot.1.6 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.17-24.nw.1.22 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.tl.1.14 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.bld.1.11 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.bll.1.11 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.brt.1.6 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.bsb.1.5 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.btl.1.8 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.btt.1.5 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.m.1.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.bh.1.6 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cf.1.5 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cu.1.9 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dr.2.10 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.du.1.12 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.em.1.6 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ev.2.5 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.ji.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.jv.2.10 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.mm.2.5 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.mq.1.6 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.om.2.10 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.pe.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.pq.1.6 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.qp.2.5 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |