Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $192$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $7 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 20 }{2}$ | ||||||
| Cusps: | $20$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{2}\cdot12^{8}\cdot24^{2}$ | Cusp orbits | $2^{10}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24AG7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.384.7.15 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&8\\0&17\end{bmatrix}$, $\begin{bmatrix}11&6\\12&1\end{bmatrix}$, $\begin{bmatrix}11&18\\0&7\end{bmatrix}$, $\begin{bmatrix}13&0\\12&13\end{bmatrix}$, $\begin{bmatrix}17&12\\12&7\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_{12}:C_2^4$ |
| Contains $-I$: | no $\quad$ (see 24.192.7.n.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $8$ |
| Full 24-torsion field degree: | $192$ |
Jacobian
| Conductor: | $2^{31}\cdot3^{7}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2^{2}$ |
| Newforms: | 24.2.a.a, 24.2.d.a, 96.2.d.a, 192.2.a.b, 192.2.a.d |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x w - z v + t u $ |
| $=$ | $x u - y z + w t$ | |
| $=$ | $y t - z w + z u$ | |
| $=$ | $x v - z w - z u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{8} z^{2} + 2 x^{6} z^{4} - 8 x^{4} y^{6} - 6 x^{4} y^{4} z^{2} - 2 x^{4} y^{2} z^{4} + \cdots + y^{8} z^{2} $ |
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 24.96.3.bn.1 :
| $\displaystyle X$ | $=$ | $\displaystyle 2z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y+v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -y+v$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{4}+X^{2}Y^{2}+Y^{3}Z+X^{2}Z^{2}-YZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.7.n.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{8}Z^{2}+2X^{6}Z^{4}-8X^{4}Y^{6}-6X^{4}Y^{4}Z^{2}-2X^{4}Y^{2}Z^{4}+X^{4}Z^{6}-8X^{2}Y^{6}Z^{2}-6X^{2}Y^{4}Z^{4}-2X^{2}Y^{2}Z^{6}+Y^{8}Z^{2} $ |
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 |
|---|---|---|---|---|---|---|
| $X_0(3)$ | $3$ | $96$ | $48$ | $0$ | $0$ | full Jacobian |
| 8.96.0-8.e.2.6 | $8$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.96.0-8.e.2.6 | $8$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 24.192.3-24.f.1.14 | $24$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 24.192.3-24.f.1.18 | $24$ | $2$ | $2$ | $3$ | $0$ | $2^{2}$ |
| 24.192.3-24.bn.1.12 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 24.192.3-24.bn.1.55 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 24.192.3-24.bq.1.12 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
| 24.192.3-24.bq.1.55 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}\cdot2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.768.13-24.j.2.13 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 24.768.13-24.j.4.9 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 24.768.13-24.l.3.7 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 24.768.13-24.l.4.5 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 24.768.13-24.bh.1.14 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 24.768.13-24.bh.3.10 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 24.768.13-24.bj.1.8 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 24.768.13-24.bj.3.7 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{3}$ |
| 24.768.17-24.dn.1.5 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2^{2}$ |
| 24.768.17-24.gb.1.9 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{6}\cdot2^{2}$ |
| 24.768.17-24.gh.1.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 24.768.17-24.gn.1.10 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 24.768.17-24.gv.5.8 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 24.768.17-24.gv.6.8 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 24.768.17-24.gv.7.8 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 24.768.17-24.gv.8.8 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 24.768.17-24.hi.5.4 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 24.768.17-24.hi.6.4 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 24.768.17-24.hi.7.4 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 24.768.17-24.hi.8.4 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{3}\cdot4$ |
| 24.768.17-24.ho.1.7 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 24.768.17-24.hq.1.11 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2^{2}$ |
| 24.768.17-24.ht.1.8 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 24.768.17-24.hw.1.12 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{6}\cdot2^{2}$ |
| 24.1152.29-24.t.2.22 | $24$ | $3$ | $3$ | $29$ | $2$ | $1^{10}\cdot2^{6}$ |