Invariants
Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $144$ | ||
Index: | $96$ | $\PSL_2$-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 | $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ | Cusp orbits | $2^{4}$ | ||
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: | 12P1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.1.1905 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&6\\6&5\end{bmatrix}$, $\begin{bmatrix}1&21\\6&19\end{bmatrix}$, $\begin{bmatrix}19&20\\6&13\end{bmatrix}$ |
$\GL_2(\Z/24\Z)$-subgroup: | Group 768.69818 |
Contains $-I$: | no $\quad$ (see 24.48.1.eq.1 for the level structure with $-I$) |
Cyclic 24-isogeny field degree: | $4$ |
Cyclic 24-torsion field degree: | $16$ |
Full 24-torsion field degree: | $768$ |
Jacobian
Conductor: | $2^{4}\cdot3^{2}$ |
Simple: | yes |
Squarefree: | yes |
Decomposition: | $1$ |
Newforms: | 144.2.a.b |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 6 x y + z^{2} $ |
$=$ | $2 x^{2} + 2 x y + 18 y^{2} - 3 z^{2} - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 3 x^{4} - 6 x^{2} y^{2} - 20 x^{2} z^{2} + 12 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 to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{3}\cdot\frac{(4z^{2}+3w^{2})(559104y^{2}z^{8}-78336y^{2}z^{6}w^{2}-196992y^{2}z^{4}w^{4}-1179360y^{2}z^{2}w^{6}-353808y^{2}w^{8}-10240z^{10}+9216z^{8}w^{2}+46656z^{6}w^{4}+212112z^{4}w^{6}+131220z^{2}w^{8}+19683w^{10})}{w^{2}z^{4}(48y^{2}z^{4}-36y^{2}z^{2}w^{2}-54y^{2}w^{4}-8z^{6}+3z^{4}w^{2})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.48.1.eq.1 :
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{6}w$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}z$ |
Equation of the image curve:
$0$ | $=$ | $ 3X^{4}-6X^{2}Y^{2}-20X^{2}Z^{2}+12Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
12.48.1-12.j.1.8 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.48.0-24.y.1.4 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.48.0-24.y.1.15 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.48.0-24.cd.1.14 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.48.0-24.cd.1.15 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.48.1-12.j.1.6 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
---|---|---|---|---|---|---|
24.288.5-24.cv.1.1 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
72.288.5-72.u.1.5 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
72.288.9-72.bm.1.8 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
72.288.9-72.bs.1.8 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
120.480.17-120.qu.1.4 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |