Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $96$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $9 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $4^{4}\cdot8^{4}\cdot12^{4}\cdot24^{4}$ | Cusp orbits | $1^{4}\cdot2^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 6$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24AH9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.384.9.125 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&20\\12&23\end{bmatrix}$, $\begin{bmatrix}5&22\\0&19\end{bmatrix}$, $\begin{bmatrix}11&0\\12&1\end{bmatrix}$, $\begin{bmatrix}17&8\\12&23\end{bmatrix}$, $\begin{bmatrix}23&6\\12&11\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_{12}:C_2^4$ |
| Contains $-I$: | no $\quad$ (see 24.192.9.bs.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: | $2^{39}\cdot3^{7}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}\cdot2^{2}$ |
| Newforms: | 24.2.a.a, 24.2.d.a, 32.2.a.a$^{2}$, 96.2.a.a, 96.2.a.b, 96.2.d.a |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x z + x w - y z $ |
| $=$ | $x w + y z + u^{2}$ | |
| $=$ | $x t - y t - y r + u v$ | |
| $=$ | $t v - t s - v r - r s$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 2 x^{6} y^{4} - 3 x^{6} y^{2} z^{2} - x^{6} z^{4} + 12 x^{4} y^{6} + 14 x^{4} y^{4} z^{2} + \cdots + 2 y^{4} z^{6} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(1:1/2:-1/3:1/6:1:0:0:1:0)$, $(-1:-1/2:1/3:-1/6:1:0:0:1:0)$, $(0:-1/2:0:-1/2:-1:0:0:1:0)$, $(0:1/2:0:1/2:-1:0:0:1:0)$ |
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 24.96.3.bn.1 :
| $\displaystyle X$ | $=$ | $\displaystyle 2u$ |
| $\displaystyle Y$ | $=$ | $\displaystyle t+r$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -t+r$ |
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 24.192.9.bs.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle u$ |
Equation of the image curve:
| $0$ | $=$ | $ -2X^{6}Y^{4}-3X^{6}Y^{2}Z^{2}-X^{6}Z^{4}+12X^{4}Y^{6}+14X^{4}Y^{4}Z^{2}+4X^{4}Y^{2}Z^{4}-16X^{2}Y^{8}-16X^{2}Y^{6}Z^{2}-14X^{2}Y^{4}Z^{4}-X^{2}Y^{2}Z^{6}+4Y^{6}Z^{4}+2Y^{4}Z^{6} $ |
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 |
| 8.96.1-8.i.2.5 | $8$ | $4$ | $4$ | $1$ | $0$ | $1^{4}\cdot2^{2}$ |
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 | $8$ | $4$ | $4$ | $1$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.192.3-24.bn.1.12 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}\cdot2$ |
| 24.192.3-24.bn.1.37 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}\cdot2$ |
| 24.192.3-24.bq.2.4 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}\cdot2$ |
| 24.192.3-24.bq.2.41 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}\cdot2$ |
| 24.192.5-24.c.1.18 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
| 24.192.5-24.c.1.40 | $24$ | $2$ | $2$ | $5$ | $0$ | $2^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.768.17-24.bm.3.1 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 24.768.17-24.bm.4.1 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 24.768.17-24.bo.1.1 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 24.768.17-24.bo.2.1 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 24.768.17-24.dh.1.22 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.768.17-24.dn.2.1 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.768.17-24.dv.1.10 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.768.17-24.dy.2.1 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.768.17-24.ez.1.7 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 24.768.17-24.ez.2.8 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 24.768.17-24.ez.3.5 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 24.768.17-24.ez.4.7 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 24.768.17-24.fd.5.5 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 24.768.17-24.fd.6.7 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 24.768.17-24.fd.7.1 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 24.768.17-24.fd.8.5 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{2}\cdot4$ |
| 24.768.17-24.fm.2.1 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.768.17-24.fp.1.9 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}$ |
| 24.768.17-24.fw.2.1 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.768.17-24.gb.1.9 | $24$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.768.17-24.iq.3.5 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 24.768.17-24.iq.4.5 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 24.768.17-24.is.1.5 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 24.768.17-24.is.2.5 | $24$ | $2$ | $2$ | $17$ | $0$ | $2^{4}$ |
| 24.1152.33-24.di.2.1 | $24$ | $3$ | $3$ | $33$ | $2$ | $1^{12}\cdot2^{6}$ |
| 48.768.25-48.dr.2.17 | $48$ | $2$ | $2$ | $25$ | $2$ | $1^{8}\cdot2^{4}$ |
| 48.768.25-48.dw.1.9 | $48$ | $2$ | $2$ | $25$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 48.768.25-48.ei.2.17 | $48$ | $2$ | $2$ | $25$ | $2$ | $1^{8}\cdot2^{4}$ |
| 48.768.25-48.el.1.9 | $48$ | $2$ | $2$ | $25$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |
| 48.768.25-48.fc.3.13 | $48$ | $2$ | $2$ | $25$ | $0$ | $4^{4}$ |
| 48.768.25-48.fc.4.13 | $48$ | $2$ | $2$ | $25$ | $0$ | $4^{4}$ |
| 48.768.25-48.fe.1.9 | $48$ | $2$ | $2$ | $25$ | $0$ | $4^{4}$ |
| 48.768.25-48.fe.2.9 | $48$ | $2$ | $2$ | $25$ | $0$ | $4^{4}$ |
| 48.768.25-48.fq.3.13 | $48$ | $2$ | $2$ | $25$ | $0$ | $4^{4}$ |
| 48.768.25-48.fq.4.13 | $48$ | $2$ | $2$ | $25$ | $0$ | $4^{4}$ |
| 48.768.25-48.fs.1.9 | $48$ | $2$ | $2$ | $25$ | $0$ | $4^{4}$ |
| 48.768.25-48.fs.2.9 | $48$ | $2$ | $2$ | $25$ | $0$ | $4^{4}$ |
| 48.768.25-48.ge.2.17 | $48$ | $2$ | $2$ | $25$ | $4$ | $1^{8}\cdot2^{4}$ |
| 48.768.25-48.gh.1.9 | $48$ | $2$ | $2$ | $25$ | $4$ | $1^{8}\cdot2^{2}\cdot4$ |
| 48.768.25-48.gm.2.17 | $48$ | $2$ | $2$ | $25$ | $2$ | $1^{8}\cdot2^{4}$ |
| 48.768.25-48.gr.1.9 | $48$ | $2$ | $2$ | $25$ | $2$ | $1^{8}\cdot2^{2}\cdot4$ |