Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $576$ | ||
| Index: | $576$ | $\PSL_2$-index: | $288$ | ||||
| Genus: | $9 = 1 + \frac{ 288 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 32 }{2}$ | ||||||
| Cusps: | $32$ (of which $2$ are rational) | Cusp widths | $6^{16}\cdot12^{16}$ | Cusp orbits | $1^{2}\cdot2^{5}\cdot4^{3}\cdot8$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 6$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.576.9.240 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&9\\12&1\end{bmatrix}$, $\begin{bmatrix}5&0\\12&19\end{bmatrix}$, $\begin{bmatrix}5&15\\12&19\end{bmatrix}$, $\begin{bmatrix}7&21\\12&13\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2^4.D_4$ |
| Contains $-I$: | no $\quad$ (see 24.288.9.bi.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $8$ |
| Full 24-torsion field degree: | $128$ |
Jacobian
| Conductor: | $2^{42}\cdot3^{17}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{2}$ |
| Newforms: | 36.2.a.a, 36.2.b.a, 192.2.a.b, 576.2.a.b$^{2}$, 576.2.a.f, 576.2.c.b |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x z + x t + z s - t s $ |
| $=$ | $x z - x w + x t - x u - u s + r s$ | |
| $=$ | $x y - x t + x u - z v + t v$ | |
| $=$ | $y s + z v + t v - t s + u s$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{6} y^{8} - 12 x^{6} y^{7} z + 24 x^{6} y^{6} z^{2} - 36 x^{6} y^{5} z^{3} + 42 x^{6} y^{4} z^{4} + \cdots - 8 y^{3} z^{11} $ |
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 |
|---|
| $(0:-1:2:1:0:-1:0:1:0)$, $(0:-1:0:1:2:1:0:1:0)$ |
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 12.144.3.c.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -3x+2v+s$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 3x+2v+s$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2v-2s$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{3}Y+XY^{3}-XZ^{3}-YZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.288.9.bi.1 :
| $\displaystyle X$ | $=$ | $\displaystyle s$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
| $0$ | $=$ | $ 3X^{6}Y^{8}-12X^{6}Y^{7}Z-8X^{4}Y^{9}Z+24X^{6}Y^{6}Z^{2}+8X^{4}Y^{8}Z^{2}+16X^{2}Y^{10}Z^{2}-36X^{6}Y^{5}Z^{3}-32X^{4}Y^{7}Z^{3}+32X^{2}Y^{9}Z^{3}-8Y^{11}Z^{3}+42X^{6}Y^{4}Z^{4}-40X^{4}Y^{6}Z^{4}+16X^{2}Y^{8}Z^{4}-40Y^{10}Z^{4}-36X^{6}Y^{3}Z^{5}+16X^{4}Y^{5}Z^{5}+32X^{2}Y^{7}Z^{5}-80Y^{9}Z^{5}+24X^{6}Y^{2}Z^{6}-40X^{4}Y^{4}Z^{6}+64X^{2}Y^{6}Z^{6}-88Y^{8}Z^{6}-12X^{6}YZ^{7}-32X^{4}Y^{3}Z^{7}+32X^{2}Y^{5}Z^{7}-80Y^{7}Z^{7}+3X^{6}Z^{8}+8X^{4}Y^{2}Z^{8}+16X^{2}Y^{4}Z^{8}-88Y^{6}Z^{8}-8X^{4}YZ^{9}+32X^{2}Y^{3}Z^{9}-80Y^{5}Z^{9}+16X^{2}Y^{2}Z^{10}-40Y^{4}Z^{10}-8Y^{3}Z^{11} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.288.3-12.c.1.7 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{4}\cdot2$ |
| 24.192.1-24.da.1.4 | $24$ | $3$ | $3$ | $1$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.192.1-24.da.3.4 | $24$ | $3$ | $3$ | $1$ | $1$ | $1^{4}\cdot2^{2}$ |
| 24.288.3-12.c.1.10 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}\cdot2$ |
| 24.288.3-24.c.1.2 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}\cdot2$ |
| 24.288.3-24.c.1.9 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}\cdot2$ |
| 24.288.5-24.gx.1.2 | $24$ | $2$ | $2$ | $5$ | $2$ | $2^{2}$ |
| 24.288.5-24.gx.1.3 | $24$ | $2$ | $2$ | $5$ | $2$ | $2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.1152.25-24.ep.1.2 | $24$ | $2$ | $2$ | $25$ | $3$ | $1^{8}\cdot2^{4}$ |
| 24.1152.25-24.eq.1.12 | $24$ | $2$ | $2$ | $25$ | $3$ | $1^{8}\cdot2^{4}$ |
| 24.1152.25-24.er.1.7 | $24$ | $2$ | $2$ | $25$ | $3$ | $1^{8}\cdot2^{4}$ |
| 24.1152.25-24.es.1.2 | $24$ | $2$ | $2$ | $25$ | $3$ | $1^{8}\cdot2^{4}$ |
| 24.1152.25-24.ff.1.8 | $24$ | $2$ | $2$ | $25$ | $4$ | $1^{8}\cdot2^{4}$ |
| 24.1152.25-24.fg.1.2 | $24$ | $2$ | $2$ | $25$ | $3$ | $1^{8}\cdot2^{4}$ |
| 24.1152.25-24.fh.1.1 | $24$ | $2$ | $2$ | $25$ | $4$ | $1^{8}\cdot2^{4}$ |
| 24.1152.25-24.fi.1.7 | $24$ | $2$ | $2$ | $25$ | $3$ | $1^{8}\cdot2^{4}$ |