Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $288$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $12^{4}\cdot24^{4}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24D9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.288.9.125 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&14\\20&17\end{bmatrix}$, $\begin{bmatrix}3&16\\8&9\end{bmatrix}$, $\begin{bmatrix}9&14\\4&15\end{bmatrix}$, $\begin{bmatrix}17&16\\16&1\end{bmatrix}$, $\begin{bmatrix}17&20\\16&17\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4.D_4^2$ |
| Contains $-I$: | no $\quad$ (see 24.144.9.df.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $256$ |
Jacobian
| Conductor: | $2^{35}\cdot3^{16}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{2}$ |
| Newforms: | 32.2.a.a, 36.2.a.a$^{2}$, 72.2.d.a, 288.2.a.a, 288.2.a.e, 288.2.d.a |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ u s + v r $ |
| $=$ | $x v + t s$ | |
| $=$ | $x u - t r$ | |
| $=$ | $y s + z v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 4 x^{6} y^{2} + x^{6} z^{2} - 8 y^{6} z^{2} + y^{4} z^{4} $ |
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 |
|---|
| $(0:0:0:1/2:0:0:1:0:0)$, $(0:0:0:-1/2:0:0:1:0:0)$ |
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.s.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -v$ |
| $\displaystyle W$ | $=$ | $\displaystyle -r$ |
Equation of the image curve:
| $0$ | $=$ | $ 4XY-ZW $ |
| $=$ | $ 2X^{3}-16Y^{3}+XZ^{2}-YW^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.144.9.df.2 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle u$ |
Equation of the image curve:
| $0$ | $=$ | $ -4X^{6}Y^{2}+X^{6}Z^{2}-8Y^{6}Z^{2}+Y^{4}Z^{4} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| $X_{\mathrm{ns}}^+(3)$ | $3$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 8.96.1-8.i.2.5 | $8$ | $3$ | $3$ | $1$ | $0$ | $1^{4}\cdot2^{2}$ |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.1-8.i.2.5 | $8$ | $3$ | $3$ | $1$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.144.4-24.s.1.32 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.4-24.s.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.4-24.z.2.1 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.4-24.z.2.48 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}\cdot2$ |
| 24.144.5-24.c.1.16 | $24$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
| 24.144.5-24.c.1.35 | $24$ | $2$ | $2$ | $5$ | $1$ | $2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.576.17-24.fk.2.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.fv.2.6 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.ib.2.5 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.im.1.1 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.kl.1.19 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.ks.2.3 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.lb.2.1 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.lg.1.10 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.lr.2.7 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.lu.2.8 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.mh.2.8 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.mi.1.7 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.nq.2.8 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.nr.2.7 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.oe.2.7 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.oh.1.8 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.os.1.11 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.ox.2.4 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.pg.2.1 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.pn.1.9 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.uq.1.7 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.va.2.8 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.wv.2.1 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.576.17-24.xh.2.5 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 48.576.21-48.ij.1.20 | $48$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2^{3}$ |
| 48.576.21-48.iq.2.6 | $48$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2\cdot4$ |
| 48.576.21-48.jd.1.20 | $48$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2^{3}$ |
| 48.576.21-48.ji.2.11 | $48$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2\cdot4$ |
| 48.576.21-48.jv.2.11 | $48$ | $2$ | $2$ | $21$ | $4$ | $1^{6}\cdot2\cdot4$ |
| 48.576.21-48.jy.1.3 | $48$ | $2$ | $2$ | $21$ | $4$ | $1^{6}\cdot2^{3}$ |
| 48.576.21-48.ki.2.6 | $48$ | $2$ | $2$ | $21$ | $2$ | $1^{6}\cdot2\cdot4$ |
| 48.576.21-48.kj.1.5 | $48$ | $2$ | $2$ | $21$ | $2$ | $1^{6}\cdot2^{3}$ |
| 48.576.21-48.kt.2.6 | $48$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2\cdot4$ |
| 48.576.21-48.ku.1.5 | $48$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2^{3}$ |
| 48.576.21-48.lg.2.11 | $48$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2\cdot4$ |
| 48.576.21-48.lj.1.3 | $48$ | $2$ | $2$ | $21$ | $3$ | $1^{6}\cdot2^{3}$ |
| 48.576.21-48.lt.1.20 | $48$ | $2$ | $2$ | $21$ | $4$ | $1^{6}\cdot2^{3}$ |
| 48.576.21-48.ly.2.11 | $48$ | $2$ | $2$ | $21$ | $4$ | $1^{6}\cdot2\cdot4$ |
| 48.576.21-48.me.1.20 | $48$ | $2$ | $2$ | $21$ | $4$ | $1^{6}\cdot2^{3}$ |
| 48.576.21-48.ml.2.6 | $48$ | $2$ | $2$ | $21$ | $4$ | $1^{6}\cdot2\cdot4$ |