Invariants
| Level: | $24$ | $\SL_2$-level: | $12$ | Newform level: | $48$ | ||
| 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: | 24.96.1.1922 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&19\\12&1\end{bmatrix}$, $\begin{bmatrix}11&16\\0&19\end{bmatrix}$, $\begin{bmatrix}11&16\\12&19\end{bmatrix}$, $\begin{bmatrix}17&9\\6&19\end{bmatrix}$, $\begin{bmatrix}17&18\\18&19\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | Group 768.335908 |
| Contains $-I$: | no $\quad$ (see 24.48.1.et.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{4}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 48.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x^{2} + 2 y z $ |
| $=$ | $8 x^{2} + 2 y^{2} - 4 y z + 18 z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 9 x^{4} + 20 x^{2} z^{2} - 2 y^{2} z^{2} + 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 to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2^3}\cdot\frac{23855104yz^{11}+16056320yz^{9}w^{2}+3760128yz^{7}w^{4}+342016yz^{5}w^{6}+8384yz^{3}w^{8}+48yzw^{10}-23592960z^{12}-14614528z^{10}w^{2}-2764800z^{8}w^{4}-114688z^{6}w^{6}+11264z^{4}w^{8}+336z^{2}w^{10}+w^{12}}{w^{2}z^{6}(2592yz^{3}+36yzw^{2}-2592z^{4}+126z^{2}w^{2}+w^{4})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.48.1.et.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 3z$ |
Equation of the image curve:
| $0$ | $=$ | $ 9X^{4}+20X^{2}Z^{2}-2Y^{2}Z^{2}+4Z^{4} $ |
Modular covers
The following modular covers realize this modular curve as a fiber product over $X(1)$.
| Factor curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 3.8.0-3.a.1.1 | $3$ | $12$ | $12$ | $0$ | $0$ | full Jacobian |
| 8.12.0.l.1 | $8$ | $8$ | $4$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.48.1-12.l.1.10 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.0-24.y.1.4 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.y.1.13 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.bx.1.11 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.bx.1.14 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.1-12.l.1.28 | $24$ | $2$ | $2$ | $1$ | $0$ | 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 |
|---|---|---|---|---|---|---|
| 24.192.3-24.dh.1.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.dv.1.16 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.ei.1.6 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.ej.1.7 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.ek.1.2 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.192.3-24.el.1.5 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.ep.1.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.3-24.et.1.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.192.5-24.bi.1.14 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.192.5-24.bj.1.16 | $24$ | $2$ | $2$ | $5$ | $3$ | $1^{4}$ |
| 24.192.5-24.bk.1.16 | $24$ | $2$ | $2$ | $5$ | $0$ | $1^{4}$ |
| 24.192.5-24.bl.1.15 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.288.5-24.ds.1.1 | $24$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 72.288.5-72.x.1.1 | $72$ | $3$ | $3$ | $5$ | $?$ | not computed |
| 72.288.9-72.bp.1.10 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 72.288.9-72.bv.1.7 | $72$ | $3$ | $3$ | $9$ | $?$ | not computed |
| 120.192.3-120.jj.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.jn.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.km.1.20 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.kn.1.26 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.ko.1.28 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.kp.1.30 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.kx.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.3-120.lb.1.32 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.192.5-120.bm.1.32 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.bn.1.32 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.bo.1.32 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.192.5-120.bp.1.32 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.480.17-120.qx.1.24 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 168.192.3-168.hj.1.32 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.hn.1.32 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.im.1.25 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.in.1.29 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.io.1.18 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ip.1.26 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.it.1.32 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.3-168.ix.1.32 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.192.5-168.bx.1.28 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.by.1.28 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.bz.1.28 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.192.5-168.ca.1.28 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.192.3-264.hj.1.32 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.hn.1.32 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.im.1.26 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.in.1.29 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.io.1.26 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ip.1.29 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.it.1.32 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.3-264.ix.1.32 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.192.5-264.bi.1.30 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.192.5-264.bj.1.30 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.192.5-264.bk.1.28 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.192.5-264.bl.1.28 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.3-312.jj.1.30 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.jn.1.30 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.km.1.29 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.kn.1.31 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.ko.1.30 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.kp.1.30 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.kx.1.30 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.3-312.lb.1.30 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.192.5-312.bm.1.26 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.bn.1.27 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.bo.1.27 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.192.5-312.bp.1.26 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |