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: | 24.288.8.2102 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&18\\12&5\end{bmatrix}$, $\begin{bmatrix}9&20\\8&9\end{bmatrix}$, $\begin{bmatrix}13&8\\8&5\end{bmatrix}$, $\begin{bmatrix}17&0\\0&5\end{bmatrix}$, $\begin{bmatrix}17&2\\16&5\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4.D_4^2$ |
| Contains $-I$: | no $\quad$ (see 24.144.8.fm.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: | $2^{30}\cdot3^{12}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}\cdot2^{2}$ |
| Newforms: | 36.2.a.a$^{3}$, 96.2.d.a$^{2}$, 144.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x z + x r + 2 y r + u v $ |
| $=$ | $x w + x v - 2 y v - t r$ | |
| $=$ | $x^{2} - 2 x y - z r - w^{2} - t^{2} + t u$ | |
| $=$ | $x^{2} + z^{2} + w^{2} + t^{2} - 2 t u + u^{2}$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 16 x^{12} + 24 x^{10} z^{2} + 48 x^{8} y^{2} z^{2} + 17 x^{8} z^{4} + 48 x^{6} y^{2} z^{4} + \cdots + y^{6} z^{6} $ |
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.w.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -z$ |
| $\displaystyle W$ | $=$ | $\displaystyle -r$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{2}-6XY+8Y^{2}-ZW+W^{2} $ |
| $=$ | $ X^{3}-2XY^{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.fm.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}r$ |
| $\displaystyle Z$ | $=$ | $\displaystyle u$ |
Equation of the image curve:
| $0$ | $=$ | $ 16X^{12}+24X^{10}Z^{2}+48X^{8}Y^{2}Z^{2}+17X^{8}Z^{4}+48X^{6}Y^{2}Z^{4}+24X^{4}Y^{4}Z^{4}+4X^{2}Y^{6}Z^{4}+6X^{6}Z^{6}+9X^{4}Y^{2}Z^{6}+6X^{2}Y^{4}Z^{6}+Y^{6}Z^{6}+X^{4}Z^{8} $ |
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.w.1.16 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 24.144.4-24.w.1.32 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
| 24.144.4-24.ch.1.48 | $24$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
| 24.144.4-24.gi.2.3 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 24.144.4-24.gi.2.30 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
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.kv.1.8 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.le.1.4 | $24$ | $2$ | $2$ | $15$ | $2$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.lw.1.8 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.lz.1.4 | $24$ | $2$ | $2$ | $15$ | $1$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.nl.1.1 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.nz.2.3 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.or.1.4 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.os.1.4 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.17-24.nh.1.24 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.tk.1.16 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.blc.1.16 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.blk.1.16 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.brs.2.6 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.bsa.1.8 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.btk.2.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.bts.1.7 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.p.1.7 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.be.1.14 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cc.1.7 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.cx.1.3 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.do.1.12 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dx.2.11 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ep.2.11 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.es.2.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jm.2.11 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.jq.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.mj.1.10 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.mt.2.11 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.oi.1.12 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.ph.2.11 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.pu.2.11 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.qm.1.10 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |