Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $72$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12P1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.1.2118 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&13\\6&19\end{bmatrix}$, $\begin{bmatrix}11&9\\18&23\end{bmatrix}$, $\begin{bmatrix}19&18\\6&11\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | Group 768.69765 |
| Contains $-I$: | no $\quad$ (see 24.48.1.ij.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{3}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 72.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} - 2 y z $ |
| $=$ | $24 x^{2} - 6 y^{2} + 12 y z - 54 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} - 20 x^{2} z^{2} + 6 y^{2} z^{2} + 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
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{2^3\cdot3^3}\cdot\frac{17390370816yz^{11}-3901685760yz^{9}w^{2}+304570368yz^{7}w^{4}-9234432yz^{5}w^{6}+75456yz^{3}w^{8}-144yzw^{10}-17199267840z^{12}+3551330304z^{10}w^{2}-223948800z^{8}w^{4}+3096576z^{6}w^{6}+101376z^{4}w^{8}-1008z^{2}w^{10}+w^{12}}{w^{2}z^{6}(23328yz^{3}-108yzw^{2}-23328z^{4}-378z^{2}w^{2}+w^{4})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.48.1.ij.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 3z$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{4}-20X^{2}Z^{2}+6Y^{2}Z^{2}+4Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.1-12.i.1.4 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.0-24.bw.1.7 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.bw.1.8 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.cb.1.6 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.cb.1.7 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.1-12.i.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.288.5-24.fr.1.4 | $24$ | $3$ | $3$ | $5$ | $2$ | $1^{4}$ |
| 72.288.5-72.bd.1.4 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.288.9-72.ci.1.4 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.288.9-72.cl.1.3 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.480.17-120.bpv.1.5 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |