Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (none of which are rational) | Cusp widths | $2^{4}\cdot6^{4}\cdot8^{2}\cdot24^{2}$ | Cusp orbits | $2^{6}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24V3 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.192.3.1745 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&10\\0&13\end{bmatrix}$, $\begin{bmatrix}1&17\\0&23\end{bmatrix}$, $\begin{bmatrix}19&9\\12&13\end{bmatrix}$, $\begin{bmatrix}23&3\\12&17\end{bmatrix}$, $\begin{bmatrix}23&12\\12&7\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2\times C_4^2:D_6$ |
| Contains $-I$: | no $\quad$ (see 24.96.3.em.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $8$ |
| Full 24-torsion field degree: | $384$ |
Jacobian
| Conductor: | $2^{12}\cdot3^{5}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{3}$ |
| Newforms: | 48.2.a.a, 144.2.a.b$^{2}$ |
Models
Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ x^{2} - x y + x z - y^{2} $ |
| $=$ | $3 x u + y u - z u + w t$ | |
| $=$ | $3 x t - 3 y t + w u$ | |
| $=$ | $x w - 5 y w + z w + t u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 1083 x^{8} - 370 x^{6} y^{2} + 3420 x^{6} z^{2} + 3 x^{4} y^{4} - 806 x^{4} y^{2} z^{2} + 2130 x^{4} z^{4} + \cdots + 75 z^{8} $ |
Geometric Weierstrass model Geometric Weierstrass model
| $ 9 w^{2} $ | $=$ | $ 9 x^{4} - 3 x^{2} z^{2} + z^{4} $ |
| $0$ | $=$ | $-3 x^{2} + y^{2} + z^{2}$ |
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 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{2^4}{11}\cdot\frac{16875968004000yzu^{10}+31975944z^{2}w^{10}+3545784z^{2}w^{8}u^{2}-466434672z^{2}w^{6}u^{4}+11810972976z^{2}w^{4}u^{6}-87381088920z^{2}w^{2}u^{8}+85471698312z^{2}t^{10}-256415094936z^{2}t^{8}u^{2}+731257863336z^{2}t^{6}u^{4}-1617631031016z^{2}t^{4}u^{6}+3940151367240z^{2}t^{2}u^{8}-6931251392412z^{2}u^{10}-107025710w^{12}-2960144w^{10}u^{2}-112638174w^{8}u^{4}+383151472w^{6}u^{6}+4231022294w^{4}u^{8}+285421761415w^{2}u^{10}+9392494320t^{12}-85471698312t^{10}u^{2}+328946023296t^{8}u^{4}-871971343056t^{6}u^{6}+2086849898496t^{4}u^{8}-2729841751980t^{2}u^{10}+376696036477u^{12}}{u^{2}(4898880yzu^{8}-15972z^{2}w^{8}-85668z^{2}w^{6}u^{2}+565620z^{2}w^{4}u^{4}-1286316z^{2}w^{2}u^{6}+2108304z^{2}t^{2}u^{6}-2012040z^{2}u^{8}+1331w^{10}+8470w^{8}u^{2}-2414500w^{6}u^{4}+3604874w^{4}u^{6}-6527625w^{2}u^{8}+2108304t^{4}u^{6}+4126200t^{2}u^{8}+109350u^{10})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.96.3.em.1 :
| $\displaystyle X$ | $=$ | $\displaystyle w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 6z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle u$ |
Equation of the image curve:
| $0$ | $=$ | $ 1083X^{8}-370X^{6}Y^{2}+3X^{4}Y^{4}+3420X^{6}Z^{2}-806X^{4}Y^{2}Z^{2}+6X^{2}Y^{4}Z^{2}+2130X^{4}Z^{4}-662X^{2}Y^{2}Z^{4}+3Y^{4}Z^{4}-900X^{2}Z^{6}-226Y^{2}Z^{6}+75Z^{8} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.96.1-12.h.1.3 | $12$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.48.0-24.bc.1.2 | $24$ | $4$ | $4$ | $0$ | $0$ | full Jacobian |
| 24.96.1-12.h.1.26 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.96.1-24.iu.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.96.1-24.iu.1.15 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.96.1-24.iu.1.18 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
| 24.96.1-24.iu.1.31 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.384.5-24.eh.1.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.eh.1.9 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.eh.2.4 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.eh.2.5 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.ei.1.3 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.ei.1.10 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.ei.2.3 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.5-24.ei.2.6 | $24$ | $2$ | $2$ | $5$ | $0$ | $2$ |
| 24.384.9-24.ks.1.7 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{6}$ |
| 24.384.9-24.kt.1.3 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{6}$ |
| 24.384.9-24.ku.1.5 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{6}$ |
| 24.384.9-24.kv.1.6 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{6}$ |
| 24.384.9-24.kw.1.4 | $24$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 24.384.9-24.kw.2.3 | $24$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 24.384.9-24.kx.1.4 | $24$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 24.384.9-24.kx.2.2 | $24$ | $2$ | $2$ | $9$ | $0$ | $2\cdot4$ |
| 24.576.13-24.hn.1.5 | $24$ | $3$ | $3$ | $13$ | $0$ | $1^{10}$ |
| 120.384.5-120.ux.1.7 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.ux.1.10 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.ux.2.4 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.ux.2.13 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.uy.1.5 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.uy.1.12 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.uy.2.8 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.5-120.uy.2.9 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.384.9-120.bko.1.5 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bkp.1.5 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bkq.1.9 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bkr.1.9 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bks.1.15 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bks.2.9 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bkt.1.15 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bkt.2.5 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.5-168.ux.1.7 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ux.1.10 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ux.2.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.ux.2.16 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.uy.1.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.uy.1.12 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.uy.2.5 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.5-168.uy.2.12 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.384.9-168.bix.1.13 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.biy.1.7 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.biz.1.11 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bja.1.11 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bjb.1.11 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bjb.2.11 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bjc.1.13 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bjc.2.7 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.5-264.ux.1.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.ux.1.10 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.ux.2.7 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.ux.2.10 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.uy.1.5 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.uy.1.12 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.uy.2.5 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.5-264.uy.2.12 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.384.9-264.bic.1.13 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bid.1.7 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bie.1.13 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bif.1.11 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.big.1.11 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.big.2.11 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bih.1.13 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bih.2.13 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.5-312.ux.1.3 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.ux.1.13 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.ux.2.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.ux.2.14 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.uy.1.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.uy.1.11 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.uy.2.5 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.5-312.uy.2.11 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.384.9-312.bko.1.5 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bkp.1.5 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bkq.1.11 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bkr.1.12 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bks.1.9 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bks.2.9 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bkt.1.5 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bkt.2.5 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |