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: | 24.48.1.218 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}13&0\\18&19\end{bmatrix}$, $\begin{bmatrix}13&12\\16&17\end{bmatrix}$, $\begin{bmatrix}15&13\\14&13\end{bmatrix}$, $\begin{bmatrix}23&5\\16&21\end{bmatrix}$ |
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.
|
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.cl.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.da.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.eo.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.1.cw.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
24.24.1.dd.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
24.24.1.en.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.dw.1 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
24.96.1.dw.2 | $24$ | $2$ | $2$ | $1$ | $1$ | dimension zero |
24.144.9.edl.1 | $24$ | $3$ | $3$ | $9$ | $3$ | $1^{8}$ |
24.192.9.py.1 | $24$ | $4$ | $4$ | $9$ | $3$ | $1^{8}$ |
120.96.1.tm.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.96.1.tm.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
120.240.17.fnd.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
120.288.17.cgvr.1 | $120$ | $6$ | $6$ | $17$ | $?$ | not computed |
168.96.1.tk.1 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
168.96.1.tk.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.96.1.tk.1 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
264.96.1.tk.2 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.96.1.tm.1 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
312.96.1.tm.2 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |