Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $96$ | ||
| Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
| Genus: | $5 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $4$ are rational) | Cusp widths | $4^{2}\cdot8^{2}\cdot12^{2}\cdot24^{2}$ | Cusp orbits | $1^{4}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
| $\overline{\Q}$-gonality: | $2 \le \gamma \le 4$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24H5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.192.5.1452 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}11&14\\12&11\end{bmatrix}$, $\begin{bmatrix}13&13\\0&7\end{bmatrix}$, $\begin{bmatrix}13&20\\0&5\end{bmatrix}$, $\begin{bmatrix}17&16\\12&13\end{bmatrix}$, $\begin{bmatrix}19&20\\12&11\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4^2:C_2^2\times S_3$ |
| Contains $-I$: | no $\quad$ (see 24.96.5.bh.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $16$ |
| Full 24-torsion field degree: | $384$ |
Jacobian
| Conductor: | $2^{24}\cdot3^{5}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot4$ |
| Newforms: | 48.2.a.a, 96.2.c.a |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ y^{2} - y z - w^{2} + w t $ |
| $=$ | $2 x^{2} + y t - z w$ | |
| $=$ | $2 y z + z^{2} + 2 w t + t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} y^{4} + 6 x^{4} y^{2} z^{2} + 9 x^{4} z^{4} + 2 x^{2} y^{5} z - 20 x^{2} y^{3} z^{3} + \cdots + 9 y^{2} z^{6} $ |
Rational points
This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:-1/2:1:-1/2:1)$, $(0:1:0:1:0)$, $(0:-1:0:1:0)$, $(0:1/2:-1:-1/2:1)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\cdot3^3\,\frac{864z^{2}w^{10}-2592z^{2}w^{9}t+4752z^{2}w^{8}t^{2}-12928z^{2}w^{7}t^{3}-1680z^{2}w^{6}t^{4}+33200z^{2}w^{5}t^{5}+15560z^{2}w^{4}t^{6}-24928z^{2}w^{3}t^{7}-29026z^{2}w^{2}t^{8}-11418z^{2}wt^{9}-1603z^{2}t^{10}-1152w^{12}+6912w^{11}t-14688w^{10}t^{2}+11808w^{9}t^{3}-3888w^{8}t^{4}-4352w^{7}t^{5}+18832w^{6}t^{6}+7632w^{5}t^{7}-12672w^{4}t^{8}-19632w^{3}t^{9}-12990w^{2}t^{10}-4006wt^{11}-451t^{12}}{128z^{2}w^{10}-384z^{2}w^{9}t-64z^{2}w^{8}t^{2}-64z^{2}w^{6}t^{4}+576z^{2}w^{5}t^{5}-488z^{2}w^{4}t^{6}-48z^{2}w^{3}t^{7}+82z^{2}w^{2}t^{8}+18z^{2}wt^{9}+z^{2}t^{10}-128w^{10}t^{2}-128w^{9}t^{3}-192w^{8}t^{4}-1024w^{7}t^{5}+2720w^{6}t^{6}-1056w^{5}t^{7}-832w^{4}t^{8}+224w^{3}t^{9}+150w^{2}t^{10}+22wt^{11}+t^{12}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.96.5.bh.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y-z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}Y^{4}+6X^{4}Y^{2}Z^{2}+9X^{4}Z^{4}+2X^{2}Y^{5}Z-20X^{2}Y^{3}Z^{3}+18X^{2}YZ^{5}+Y^{6}Z^{2}-10Y^{4}Z^{4}+9Y^{2}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.96.1-12.h.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | $4$ |
| 24.96.1-12.h.1.23 | $24$ | $2$ | $2$ | $1$ | $0$ | $4$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.384.9-24.hv.1.1 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 24.384.9-24.ib.1.6 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 24.384.9-24.jf.2.3 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{2}\cdot2$ |
| 24.384.9-24.jl.2.4 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 24.384.9-24.ju.2.6 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 24.384.9-24.ju.2.12 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 24.384.9-24.jy.2.10 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{4}$ |
| 24.384.9-24.jy.2.15 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{4}$ |
| 24.384.9-24.kd.2.9 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 24.384.9-24.kd.2.16 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 24.384.9-24.kf.2.3 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 24.384.9-24.kf.2.16 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 24.384.9-24.kg.2.8 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 24.384.9-24.kg.2.11 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 24.384.9-24.kk.1.10 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 24.384.9-24.kk.1.15 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{4}$ |
| 24.384.9-24.kp.1.9 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 24.384.9-24.kp.1.16 | $24$ | $2$ | $2$ | $9$ | $2$ | $1^{4}$ |
| 24.384.9-24.kr.2.7 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 24.384.9-24.kr.2.14 | $24$ | $2$ | $2$ | $9$ | $1$ | $1^{2}\cdot2$ |
| 24.384.9-24.kx.2.3 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 24.384.9-24.ld.2.4 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 24.384.9-24.lr.1.3 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 24.384.9-24.lx.1.2 | $24$ | $2$ | $2$ | $9$ | $0$ | $1^{2}\cdot2$ |
| 24.576.17-24.ctq.1.13 | $24$ | $3$ | $3$ | $17$ | $0$ | $1^{4}\cdot4^{2}$ |
| 120.384.9-120.bav.2.13 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bbh.2.15 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bcj.1.5 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bcv.1.7 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bfp.1.16 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bfp.1.21 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bfq.1.10 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bfq.1.31 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bfs.1.12 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bfs.1.29 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bft.1.15 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bft.1.26 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bgb.1.11 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bgb.1.24 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bgc.2.18 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bgc.2.31 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bge.2.20 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bge.2.29 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bgf.1.14 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bgf.1.23 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bhx.1.11 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bij.1.3 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bjl.2.9 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 120.384.9-120.bjx.2.11 | $120$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bak.1.3 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.baw.1.15 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bby.1.15 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bck.2.9 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bfe.2.15 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bfe.2.20 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bff.2.15 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bff.2.22 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bfh.2.12 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bfh.2.29 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bfi.2.14 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bfi.2.23 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bfq.2.13 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bfq.2.24 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bfr.1.16 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bfr.1.21 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bft.1.11 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bft.1.30 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bfu.2.11 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bfu.2.30 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bgg.2.13 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bgs.2.7 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.bhu.1.13 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 168.384.9-168.big.1.6 | $168$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.zp.2.5 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bab.2.15 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bbd.2.5 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bbp.2.15 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bej.2.12 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bej.2.23 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bek.2.11 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bek.2.30 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bem.2.9 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bem.2.32 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.ben.2.7 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.ben.2.30 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bev.2.15 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bev.2.20 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bew.2.10 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bew.2.31 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bey.2.9 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bey.2.32 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bez.2.7 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bez.2.30 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bfl.1.11 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bfx.1.6 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bgz.1.11 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 264.384.9-264.bhl.1.6 | $264$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bav.1.13 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bbh.1.10 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bcj.1.5 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bcv.1.5 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bfp.2.7 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bfp.2.25 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bfq.2.6 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bfq.2.25 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bfs.2.10 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bfs.2.21 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bft.2.15 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bft.2.17 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bgb.2.5 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bgb.2.26 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bgc.1.6 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bgc.1.28 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bge.2.11 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bge.2.24 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bgf.1.15 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bgf.1.20 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bhx.1.5 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bij.1.5 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bjl.1.10 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |
| 312.384.9-312.bjx.1.13 | $312$ | $2$ | $2$ | $9$ | $?$ | not computed |