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: | $10$ (of which $2$ are rational) | Cusp widths | $6^{4}\cdot12^{2}\cdot24^{4}$ | Cusp orbits | $1^{2}\cdot2^{2}\cdot4$ | ||
| 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: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24D8 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.288.8.2106 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&20\\16&13\end{bmatrix}$, $\begin{bmatrix}7&14\\8&17\end{bmatrix}$, $\begin{bmatrix}19&0\\0&5\end{bmatrix}$, $\begin{bmatrix}19&14\\16&1\end{bmatrix}$, $\begin{bmatrix}23&16\\8&5\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4.D_4^2$ |
| Contains $-I$: | no $\quad$ (see 24.144.8.fi.2 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: | $2^{26}\cdot3^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{4}\cdot2^{2}$ |
| Newforms: | 36.2.a.a$^{3}$, 72.2.d.b, 144.2.a.a, 288.2.d.b |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x u + x r + 2 y^{2} - y u + y r - w v $ |
| $=$ | $x u + x r - 2 y u - w t$ | |
| $=$ | $2 y^{2} - y u - y r + z w - w t + w v$ | |
| $=$ | $2 x t - 2 y t - z u + z r - w r$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{12} + 2 x^{10} y^{2} - 3 x^{8} y^{4} - 8 x^{6} y^{6} - 36 x^{6} y^{4} z^{2} - 36 x^{4} y^{6} z^{2} + \cdots + 108 y^{6} z^{6} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:0:-1:1:0:1)$, $(0:0:0:0:1:1:0:1)$ |
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 :
| $\displaystyle X$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle u$ |
| $\displaystyle W$ | $=$ | $\displaystyle r$ |
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.fi.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}t$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{12}+2X^{10}Y^{2}-3X^{8}Y^{4}-8X^{6}Y^{6}-36X^{6}Y^{4}Z^{2}-36X^{4}Y^{6}Z^{2}-108X^{4}Y^{4}Z^{4}+8X^{2}Y^{10}+36X^{2}Y^{8}Z^{2}-108X^{2}Y^{4}Z^{6}+4Y^{12}+36Y^{10}Z^{2}+108Y^{8}Z^{4}+108Y^{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.u.1.15 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 24.144.4-24.u.1.25 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 24.144.4-24.x.2.11 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 24.144.4-24.x.2.27 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{2}\cdot2$ |
| 24.144.4-24.ch.1.19 | $24$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $2^{2}$ |
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.kx.1.7 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.ld.2.5 | $24$ | $2$ | $2$ | $15$ | $2$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.lv.1.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.mb.2.5 | $24$ | $2$ | $2$ | $15$ | $1$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.np.2.10 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.nw.1.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.oo.1.7 | $24$ | $2$ | $2$ | $15$ | $0$ | $1^{3}\cdot2^{2}$ |
| 24.576.15-24.ou.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.nw.2.2 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.ti.2.3 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.tl.1.14 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.blb.2.2 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.blc.1.16 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.blj.1.14 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.blk.2.3 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.brp.1.3 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.bru.2.6 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.brx.2.5 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.bsc.1.4 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.btg.1.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.btn.1.3 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.bto.1.8 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 24.576.17-24.btv.2.7 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.l.1.14 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.bg.2.13 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ce.2.13 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.ct.1.13 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dq.1.7 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.dt.1.16 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.el.2.6 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{5}\cdot2^{2}$ |
| 48.576.17-48.eu.1.3 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{5}\cdot2^{2}$ |
| 48.576.19-48.jg.1.16 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.jt.1.7 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.ml.1.3 | $48$ | $2$ | $2$ | $19$ | $0$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.mp.2.6 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.ol.1.6 | $48$ | $2$ | $2$ | $19$ | $2$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.pd.1.15 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.pn.2.3 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |
| 48.576.19-48.qo.1.2 | $48$ | $2$ | $2$ | $19$ | $1$ | $1^{5}\cdot2\cdot4$ |