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: | $2$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8F1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.1.181 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}11&15\\2&13\end{bmatrix}$, $\begin{bmatrix}11&16\\4&15\end{bmatrix}$, $\begin{bmatrix}19&18\\2&17\end{bmatrix}$, $\begin{bmatrix}21&11\\16&11\end{bmatrix}$ |
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 $ | $=$ | $ 3 x^{2} + y^{2} - y z + z^{2} + 2 w^{2} $ |
$=$ | $y^{2} + z^{2} - 4 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 9 x^{4} - 4 x^{2} z^{2} + y^{4} + 4 y^{2} z^{2} + 4 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:
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle 3x$ |
$\displaystyle Z$ | $=$ | $\displaystyle 3w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{z^{3}(z-2w)(z+2w)(12yz^{4}w^{2}-48yz^{2}w^{4}-64yw^{6}+z^{7}-8z^{5}w^{2}-32z^{3}w^{4}+192zw^{6})}{w^{8}(4yzw^{2}+z^{4}-4z^{2}w^{2}-4w^{4})}$ |
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.bs.1 | $8$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.co.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.cv.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.eu.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.1.di.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.24.1.dy.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.24.1.el.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 |
---|---|---|---|---|---|---|
24.96.3.ig.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
24.96.3.ih.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
24.144.9.ege.1 | $24$ | $3$ | $3$ | $9$ | $4$ | $1^{8}$ |
24.192.9.rd.1 | $24$ | $4$ | $4$ | $9$ | $1$ | $1^{8}$ |
48.96.3.qz.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
48.96.3.rb.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
48.96.3.un.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
48.96.3.up.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
48.96.5.kb.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
48.96.5.kj.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{2}\cdot2$ |
48.96.5.uj.1 | $48$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
48.96.5.ur.1 | $48$ | $2$ | $2$ | $5$ | $3$ | $1^{2}\cdot2$ |
120.96.3.vc.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.96.3.vd.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
120.240.17.fpw.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
120.288.17.cgzy.1 | $120$ | $6$ | $6$ | $17$ | $?$ | not computed |
168.96.3.so.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
168.96.3.sp.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.dzf.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.dzh.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.eap.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.3.ear.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
240.96.5.cbp.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.cbt.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.ctd.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.96.5.cth.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
264.96.3.so.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
264.96.3.sp.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.96.3.vc.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
312.96.3.vd.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |