Invariants
Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $288$ | ||
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 | $4^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8G1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.1.247 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&8\\12&19\end{bmatrix}$, $\begin{bmatrix}7&19\\2&7\end{bmatrix}$, $\begin{bmatrix}11&19\\2&23\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | none in database |
Cyclic 24-isogeny field degree: | $16$ |
Cyclic 24-torsion field degree: | $128$ |
Full 24-torsion field degree: | $1536$ |
Jacobian
Conductor: | $2^{5}\cdot3^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 288.2.a.d |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 4 x y + z^{2} + z w + w^{2} $ |
$=$ | $16 x^{2} + 2 y^{2} - z^{2} + 2 z w + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} + 2 x^{3} z - x^{2} y^{2} + 3 x^{2} z^{2} + 2 x y^{2} z + 2 x z^{3} + 2 y^{4} + 2 y^{2} z^{2} + z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle y$ |
$\displaystyle Z$ | $=$ | $\displaystyle w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -2^4\cdot3^3\,\frac{34y^{2}z^{10}+236y^{2}z^{9}w+252y^{2}z^{8}w^{2}-1248y^{2}z^{7}w^{3}-3768y^{2}z^{6}w^{4}-3600y^{2}z^{5}w^{5}-864y^{2}z^{4}w^{6}+384y^{2}z^{3}w^{7}+216y^{2}z^{2}w^{8}+80y^{2}zw^{9}+16y^{2}w^{10}-11z^{12}-192z^{11}w-978z^{10}w^{2}-2484z^{9}w^{3}-3666z^{8}w^{4}-3168z^{7}w^{5}-1048z^{6}w^{6}+912z^{5}w^{7}+1416z^{4}w^{8}+880z^{3}w^{9}+264z^{2}w^{10}+48zw^{11}+8w^{12}}{(z^{2}+zw+w^{2})^{4}(2y^{2}z^{2}-4y^{2}zw-4y^{2}w^{2}+z^{4}+8z^{3}w+6z^{2}w^{2}-4zw^{3}-2w^{4})}$ |
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.bk.2 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.em.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.1.ea.1 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
---|---|---|---|---|---|---|
24.144.9.edu.1 | $24$ | $3$ | $3$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
24.192.9.qh.1 | $24$ | $4$ | $4$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
48.96.5.hj.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
48.96.5.hl.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.96.5.jf.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.96.5.jh.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
48.96.5.rr.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
48.96.5.rt.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.96.5.tn.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.96.5.tp.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
120.240.17.fnm.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
120.288.17.cgwa.2 | $120$ | $6$ | $6$ | $17$ | $?$ | not computed |
240.96.5.byd.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.byf.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.byl.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.byn.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.cpr.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.cpt.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.cpz.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.cqb.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |