Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $96$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $2^{4}\cdot4^{6}\cdot6^{4}\cdot8^{2}\cdot12^{6}\cdot24^{2}$ | Cusp orbits | $2^{8}\cdot4^{2}$ | ||
| 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: | 24AB5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.384.5.7424 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&10\\12&17\end{bmatrix}$, $\begin{bmatrix}7&2\\0&1\end{bmatrix}$, $\begin{bmatrix}11&20\\12&1\end{bmatrix}$, $\begin{bmatrix}13&18\\12&23\end{bmatrix}$, $\begin{bmatrix}23&20\\12&23\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_{12}:C_2^4$ |
| Contains $-I$: | no $\quad$ (see 24.192.5.cq.4 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: | $2^{19}\cdot3^{5}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2^{2}$ |
| Newforms: | 24.2.a.a, 24.2.d.a, 96.2.f.a |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 2 x^{2} - 2 x w - y t $ |
| $=$ | $2 x^{2} + 2 x w + y^{2} - y t + 2 w^{2} - t^{2}$ | |
| $=$ | $2 x^{2} - y^{2} + 4 z^{2} - 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 6 x^{6} y^{2} + 8 x^{4} y^{4} - 16 x^{4} y^{2} z^{2} + x^{4} z^{4} + 8 x^{2} y^{6} + \cdots + 4 y^{4} z^{4} $ |
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.bq.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y+2z+t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -y+2z+t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle y+2z-t$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{3}Y-X^{2}Y^{2}-X^{3}Z+X^{2}YZ-Y^{3}Z+2X^{2}Z^{2}-3XYZ^{2}-Y^{2}Z^{2}-XZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.5.cq.4 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ -6X^{6}Y^{2}+8X^{4}Y^{4}-16X^{4}Y^{2}Z^{2}+X^{4}Z^{4}+8X^{2}Y^{6}-4X^{2}Y^{2}Z^{4}+4Y^{4}Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.192.1-24.ck.3.16 | $24$ | $2$ | $2$ | $1$ | $0$ | $2^{2}$ |
| 24.192.1-24.ck.3.28 | $24$ | $2$ | $2$ | $1$ | $0$ | $2^{2}$ |
| 24.192.3-24.bq.2.21 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.bq.2.47 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.br.1.23 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.br.1.45 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.768.13-24.p.7.14 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.x.3.13 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.bf.2.13 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.bj.4.15 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.bv.3.16 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.bz.3.15 | $24$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.cd.4.16 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.ch.2.16 | $24$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.cs.1.16 | $24$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.cx.5.10 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.da.1.16 | $24$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.df.4.10 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.dy.3.12 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.ed.7.9 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.ef.4.14 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.ep.4.9 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.768.17-24.ez.4.2 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}\cdot4$ |
| 24.768.17-24.hk.2.8 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}\cdot4$ |
| 24.768.17-24.ix.2.1 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}\cdot4$ |
| 24.768.17-24.jq.1.7 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}\cdot4$ |
| 24.768.17-24.lc.2.10 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}\cdot4$ |
| 24.768.17-24.lt.2.12 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}\cdot4$ |
| 24.768.17-24.me.1.9 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}\cdot4$ |
| 24.768.17-24.mp.1.11 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}\cdot4$ |
| 24.1152.25-24.t.2.25 | $24$ | $3$ | $3$ | $25$ | $0$ | $1^{4}\cdot2^{6}\cdot4$ |