Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $576$ | ||
| 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 | $2^{8}\cdot6^{8}\cdot8^{4}\cdot24^{4}$ | Cusp orbits | $2^{6}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24Z5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.384.5.2831 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&22\\0&13\end{bmatrix}$, $\begin{bmatrix}11&0\\0&7\end{bmatrix}$, $\begin{bmatrix}19&2\\0&1\end{bmatrix}$, $\begin{bmatrix}23&1\\0&13\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $D_8:D_6$ |
| Contains $-I$: | no $\quad$ (see 24.192.5.ey.3 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $1$ |
| Cyclic 24-torsion field degree: | $8$ |
| Full 24-torsion field degree: | $192$ |
Jacobian
| Conductor: | $2^{23}\cdot3^{7}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{3}\cdot2$ |
| Newforms: | 24.2.a.a, 48.2.c.a, 576.2.a.b, 576.2.a.d |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 3 x^{2} - 2 x y - y^{2} - z^{2} $ |
| $=$ | $3 x^{2} + 4 x y - y^{2} - z^{2} - w^{2}$ | |
| $=$ | $6 z^{2} + 4 w^{2} - t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3600 x^{8} + 504 x^{6} y^{2} - 3360 x^{6} z^{2} + 9 x^{4} y^{4} - 120 x^{4} y^{2} z^{2} + 184 x^{4} z^{4} + \cdots + 25 z^{8} $ |
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 192 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{(2w^{2}-t^{2})^{3}(559104y^{2}w^{16}+104448y^{2}w^{14}t^{2}-376320y^{2}w^{12}t^{4}+2970624y^{2}w^{10}t^{6}-3236736y^{2}w^{8}t^{8}+1642368y^{2}w^{6}t^{10}-463008y^{2}w^{4}t^{12}+69888y^{2}w^{2}t^{14}-4368y^{2}t^{16}-280064w^{18}+40704w^{16}t^{2}-158976w^{14}t^{4}+502656w^{12}t^{6}-425472w^{10}t^{8}+166464w^{8}t^{10}-31248w^{6}t^{12}+2328w^{4}t^{14}-18w^{2}t^{16}+t^{18})}{t^{2}w^{8}(2w-t)(2w+t)(96y^{2}w^{10}+48y^{2}w^{8}t^{2}-264y^{2}w^{6}t^{4}+204y^{2}w^{4}t^{6}-60y^{2}w^{2}t^{8}+6y^{2}t^{10}+16w^{12}+12w^{10}t^{2}+141w^{8}t^{4}-136w^{6}t^{6}+57w^{4}t^{8}-12w^{2}t^{10}+t^{12})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.5.ey.3 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2z+2t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 3600X^{8}+504X^{6}Y^{2}+9X^{4}Y^{4}-3360X^{6}Z^{2}-120X^{4}Y^{2}Z^{2}+184X^{4}Z^{4}-42X^{2}Y^{2}Z^{4}+280X^{2}Z^{6}+25Z^{8} $ |
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.da.1.4 | $24$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 24.192.1-24.da.1.5 | $24$ | $2$ | $2$ | $1$ | $1$ | $1^{2}\cdot2$ |
| 24.192.1-24.de.3.13 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.de.3.19 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.dm.2.3 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.1-24.dm.2.5 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}\cdot2$ |
| 24.192.3-24.fb.1.6 | $24$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 24.192.3-24.fb.1.23 | $24$ | $2$ | $2$ | $3$ | $1$ | $2$ |
| 24.192.3-24.fq.1.4 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.fq.1.9 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.gl.3.9 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gl.3.15 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gt.3.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.gt.3.10 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
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.fs.4.2 | $24$ | $2$ | $2$ | $13$ | $1$ | $2^{4}$ |
| 24.768.13-24.ft.1.1 | $24$ | $2$ | $2$ | $13$ | $1$ | $2^{4}$ |
| 24.768.13-24.fu.1.3 | $24$ | $2$ | $2$ | $13$ | $1$ | $2^{4}$ |
| 24.768.13-24.fv.3.6 | $24$ | $2$ | $2$ | $13$ | $1$ | $2^{4}$ |
| 24.1152.25-24.es.1.2 | $24$ | $3$ | $3$ | $25$ | $3$ | $1^{10}\cdot2^{5}$ |
| 48.768.13-48.kn.5.2 | $48$ | $2$ | $2$ | $13$ | $1$ | $2^{4}$ |
| 48.768.13-48.kn.6.2 | $48$ | $2$ | $2$ | $13$ | $1$ | $2^{4}$ |
| 48.768.13-48.ko.5.2 | $48$ | $2$ | $2$ | $13$ | $1$ | $2^{4}$ |
| 48.768.13-48.ko.6.2 | $48$ | $2$ | $2$ | $13$ | $1$ | $2^{4}$ |
| 48.768.17-48.wk.2.14 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2\cdot4$ |
| 48.768.17-48.wt.2.14 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2\cdot4$ |
| 48.768.17-48.ble.1.7 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{6}\cdot2\cdot4$ |
| 48.768.17-48.bln.1.7 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{6}\cdot2\cdot4$ |