Invariants
Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
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: | 8F1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.1.473 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&8\\10&15\end{bmatrix}$, $\begin{bmatrix}7&7\\14&9\end{bmatrix}$, $\begin{bmatrix}11&17\\4&5\end{bmatrix}$ |
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: | $2^{5}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 6 x^{2} - y^{2} + y z - z^{2} $ |
$=$ | $6 x^{2} + 6 y^{2} - 9 y z + 6 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 4 x^{4} - 20 x^{2} y^{2} - 6 x^{2} z^{2} + 49 y^{4} + 42 y^{2} z^{2} + 9 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:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle 2x$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{2}{3}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^8}{3^2}\cdot\frac{471801510yz^{11}+1981474488yz^{9}w^{2}+1768761036yz^{7}w^{4}-48231288yz^{5}w^{6}-160641306yz^{3}w^{8}-60505200yzw^{10}-316758519z^{12}-377270622z^{10}w^{2}+1960700175z^{8}w^{4}+3223754100z^{6}w^{6}+1369211067z^{4}w^{8}+175681170z^{2}w^{10}-2100875w^{12}}{7766280yz^{11}-3882060yz^{9}w^{2}-12674340yz^{7}w^{4}-2765952yz^{5}w^{6}+3764768yz^{3}w^{8}+537824yzw^{10}-5214132z^{12}-8432856z^{10}w^{2}+7286517z^{8}w^{4}+8598324z^{6}w^{6}-48020z^{4}w^{8}-499408z^{2}w^{10}-268912w^{12}}$ |
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 |
---|---|---|---|---|---|---|
8.24.1.p.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.24.0.x.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.bb.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.et.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.eu.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.1.bk.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.24.1.bn.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.bad.1 | $24$ | $3$ | $3$ | $9$ | $3$ | $1^{8}$ |
24.192.9.jt.1 | $24$ | $4$ | $4$ | $9$ | $2$ | $1^{8}$ |
48.96.5.gg.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
48.96.5.gi.1 | $48$ | $2$ | $2$ | $5$ | $2$ | $1^{2}\cdot2$ |
48.96.5.gk.1 | $48$ | $2$ | $2$ | $5$ | $0$ | $1^{2}\cdot2$ |
48.96.5.gm.1 | $48$ | $2$ | $2$ | $5$ | $4$ | $1^{2}\cdot2$ |
120.240.17.qf.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
120.288.17.ron.1 | $120$ | $6$ | $6$ | $17$ | $?$ | not computed |
240.96.5.ra.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.rc.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.re.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.rg.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |