Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $96$ | ||
| Index: | $384$ | $\PSL_2$-index: | $192$ | ||||
| Genus: | $5 = 1 + \frac{ 192 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 24 }{2}$ | ||||||
| Cusps: | $24$ (none of which are rational) | Cusp widths | $1^{4}\cdot2^{2}\cdot3^{4}\cdot4^{2}\cdot6^{2}\cdot8^{4}\cdot12^{2}\cdot24^{4}$ | Cusp orbits | $2^{6}\cdot4^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24AA5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.384.5.6145 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&9\\0&5\end{bmatrix}$, $\begin{bmatrix}1&12\\0&5\end{bmatrix}$, $\begin{bmatrix}11&16\\0&1\end{bmatrix}$, $\begin{bmatrix}17&9\\0&5\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $D_8:D_6$ |
| Contains $-I$: | no $\quad$ (see 24.192.5.fz.3 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $1$ |
| Cyclic 24-torsion field degree: | $4$ |
| Full 24-torsion field degree: | $192$ |
Jacobian
| Conductor: | $2^{19}\cdot3^{5}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1\cdot2^{2}$ |
| Newforms: | 24.2.a.a, 24.2.f.a, 96.2.d.a |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ 2 x^{2} - 2 x z + 2 y^{2} + w t $ |
| $=$ | $2 x^{2} + 4 x z + 2 y^{2} + w^{2}$ | |
| $=$ | $4 x^{2} + 2 x z - 2 y^{2} - 6 z^{2} + w t + t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 36 x^{4} y^{4} + 12 x^{4} y^{2} z^{2} + x^{4} z^{4} + 16 x^{2} y^{2} z^{4} - 24 y^{6} z^{2} + \cdots + 2 y^{2} z^{6} $ |
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
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 24.96.3.ge.3 :
| $\displaystyle X$ | $=$ | $\displaystyle 3y-w+t$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 3y+w-t$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -2w-t$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{3}Y+2X^{2}Y^{2}+XY^{3}+2X^{2}YZ-2XY^{2}Z+2XYZ^{2}+XZ^{3}-YZ^{3} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.192.5.fz.3 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ 36X^{4}Y^{4}+12X^{4}Y^{2}Z^{2}+X^{4}Z^{4}+16X^{2}Y^{2}Z^{4}-24Y^{6}Z^{2}+8Y^{4}Z^{4}+2Y^{2}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.192.1-24.de.3.10 | $24$ | $2$ | $2$ | $1$ | $0$ | $2^{2}$ |
| 24.192.1-24.de.3.16 | $24$ | $2$ | $2$ | $1$ | $0$ | $2^{2}$ |
| 24.192.3-24.ge.3.20 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.ge.3.26 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gf.2.3 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
| 24.192.3-24.gf.2.5 | $24$ | $2$ | $2$ | $3$ | $0$ | $2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.768.13-24.do.1.1 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.er.2.8 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.fb.7.4 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.fc.3.8 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.fm.1.1 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.fs.4.8 | $24$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.gc.7.4 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 24.768.13-24.ge.7.8 | $24$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 24.1152.25-24.ed.1.1 | $24$ | $3$ | $3$ | $25$ | $0$ | $1^{4}\cdot2^{6}\cdot4$ |
| 48.768.13-48.lt.3.7 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.lz.3.7 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.mx.1.8 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.nh.1.8 | $48$ | $2$ | $2$ | $13$ | $1$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.oh.4.5 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.oj.4.5 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.ph.3.7 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.13-48.pv.3.7 | $48$ | $2$ | $2$ | $13$ | $0$ | $1^{2}\cdot2^{3}$ |
| 48.768.17-48.bbr.3.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}\cdot4$ |
| 48.768.17-48.bcf.3.10 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}\cdot4$ |
| 48.768.17-48.bdd.4.12 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}\cdot4$ |
| 48.768.17-48.bdf.4.12 | $48$ | $2$ | $2$ | $17$ | $0$ | $1^{4}\cdot2^{2}\cdot4$ |
| 48.768.17-48.bin.1.9 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}\cdot4$ |
| 48.768.17-48.bix.1.9 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}\cdot4$ |
| 48.768.17-48.bjv.3.10 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{4}\cdot2^{2}\cdot4$ |
| 48.768.17-48.bkb.3.10 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{4}\cdot2^{2}\cdot4$ |