Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $576$ | ||
| Index: | $48$ | $\PSL_2$-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 | $4^{4}\cdot8^{4}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $1$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.1.219 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&8\\22&1\end{bmatrix}$, $\begin{bmatrix}11&17\\6&17\end{bmatrix}$, $\begin{bmatrix}15&13\\16&13\end{bmatrix}$, $\begin{bmatrix}19&4\\0&19\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $64$ |
| Full 24-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{6}\cdot3^{2}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 576.2.a.c |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x y - y z - z^{2} $ |
| $=$ | $16 x^{2} - 2 x y - y^{2} - 3 y z - 3 z^{2} + 6 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 4 x^{3} z - 6 x^{2} y^{2} - 8 x z^{3} - 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 between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{22960xz^{11}+51936xz^{9}w^{2}+43200xz^{7}w^{4}+16448xz^{5}w^{6}+2832xz^{3}w^{8}+4060y^{2}z^{10}+10455y^{2}z^{8}w^{2}+10764y^{2}z^{6}w^{4}+6000y^{2}z^{4}w^{6}+2268y^{2}z^{2}w^{8}+585y^{2}w^{10}+21980yz^{11}+58356yz^{9}w^{2}+62856yz^{7}w^{4}+36608yz^{5}w^{6}+12900yz^{3}w^{8}+2340yzw^{10}+19602z^{12}+41832z^{10}w^{2}+31596z^{8}w^{4}+10304z^{6}w^{6}+1698z^{4}w^{8}+216z^{2}w^{10}-54w^{12}}{z^{4}(22960xz^{7}-18912xz^{5}w^{2}+8352xz^{3}w^{4}-1728xzw^{6}+4060y^{2}z^{6}-2091y^{2}z^{4}w^{2}+684y^{2}z^{2}w^{4}-108y^{2}w^{6}+21980yz^{7}-9540yz^{5}w^{2}+3024yz^{3}w^{4}-432yzw^{6}+19602z^{8}-18720z^{6}w^{2}+9882z^{4}w^{4}-3240z^{2}w^{6}+648w^{8})}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.0.bf.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0.ch.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0.cq.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0.ez.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.1.cx.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.24.1.dn.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.24.1.ej.1 | $24$ | $2$ | $2$ | $1$ | $1$ | 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 |
|---|---|---|---|---|---|---|
| 24.96.1.du.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.96.1.du.2 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
| 24.144.9.ecv.1 | $24$ | $3$ | $3$ | $9$ | $3$ | $1^{8}$ |
| 24.192.9.pi.1 | $24$ | $4$ | $4$ | $9$ | $3$ | $1^{8}$ |
| 120.96.1.tk.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.96.1.tk.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.240.17.fmn.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 120.288.17.cgvb.1 | $120$ | $6$ | $6$ | $17$ | $?$ | not computed |
| 168.96.1.ti.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.96.1.ti.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1.ti.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.96.1.ti.2 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1.tk.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.96.1.tk.2 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |