Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$ | ||||||
| Cusps: | $12$ (of which $8$ are rational) | Cusp widths | $6^{8}\cdot24^{4}$ | Cusp orbits | $1^{8}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $8$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24J7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.288.7.643 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&2\\8&9\end{bmatrix}$, $\begin{bmatrix}7&12\\0&1\end{bmatrix}$, $\begin{bmatrix}9&20\\16&21\end{bmatrix}$, $\begin{bmatrix}11&0\\0&5\end{bmatrix}$, $\begin{bmatrix}23&18\\0&5\end{bmatrix}$, $\begin{bmatrix}23&18\\0&17\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2^2\times D_4^2$ |
| Contains $-I$: | no $\quad$ (see 24.144.7.ih.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $2$ |
| Cyclic 24-torsion field degree: | $16$ |
| Full 24-torsion field degree: | $256$ |
Jacobian
| Conductor: | $2^{20}\cdot3^{11}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{7}$ |
| Newforms: | 24.2.a.a$^{2}$, 36.2.a.a$^{3}$, 48.2.a.a, 144.2.a.a |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x y - u v $ |
| $=$ | $x z - t v$ | |
| $=$ | $x z - w u$ | |
| $=$ | $x^{2} - x y + w t$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} y z^{2} + x^{4} y^{4} - x^{4} z^{4} - 6 x^{3} y^{3} z^{2} - 2 x^{2} y^{6} + x y^{5} z^{2} + y^{8} $ |
Rational points
This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:0:1:0:0)$, $(0:1/2:-1/2:-1:0:0:1)$, $(0:0:0:1:0:0:0)$, $(0:0:0:0:0:1:0)$, $(0:0:0:0:0:0:1)$, $(0:1/2:1/2:0:1:1:0)$, $(0:-1/2:1/2:-1:0:0:1)$, $(0:-1/2:-1/2:0:1:1:0)$ |
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 24.72.4.q.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle t-v$ |
| $\displaystyle W$ | $=$ | $\displaystyle -w-u$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{2}-4Y^{2}-ZW $ |
| $=$ | $ 3X^{3}-YZ^{2}-2XZW+YW^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.144.7.ih.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle w$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{5}YZ^{2}+X^{4}Y^{4}-X^{4}Z^{4}-6X^{3}Y^{3}Z^{2}-2X^{2}Y^{6}+XY^{5}Z^{2}+Y^{8} $ |
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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}^+(3)$ | $3$ | $48$ | $24$ | $0$ | $0$ | full Jacobian |
| 8.48.0-8.i.1.2 | $8$ | $6$ | $6$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.144.3-12.k.1.8 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.3-12.k.1.16 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.3-24.or.1.17 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.3-24.or.1.31 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.3-24.po.1.11 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.3-24.po.1.19 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{4}$ |
| 24.144.4-24.q.1.5 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.q.1.22 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.ch.1.21 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.gt.1.10 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.gt.1.19 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.hq.1.6 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
| 24.144.4-24.hq.1.11 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{3}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.576.13-24.ec.1.2 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{6}$ |
| 24.576.13-24.ed.1.1 | $24$ | $2$ | $2$ | $13$ | $2$ | $1^{6}$ |
| 24.576.13-24.ek.1.1 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{6}$ |
| 24.576.13-24.el.1.1 | $24$ | $2$ | $2$ | $13$ | $1$ | $1^{6}$ |
| 24.576.15-24.nl.1.1 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.nl.2.9 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.nm.1.1 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.nm.2.3 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.nn.1.9 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.nn.2.1 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.no.1.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.no.2.9 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.np.1.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.np.2.10 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.nq.1.3 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.nq.2.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.nr.1.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.nr.2.15 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.ns.1.3 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.ns.2.3 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.nt.1.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.15-24.nt.2.5 | $24$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 24.576.17-24.hs.1.36 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
| 24.576.17-24.jy.1.16 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
| 24.576.17-24.bmr.1.19 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
| 24.576.17-24.bms.1.15 | $24$ | $2$ | $2$ | $17$ | $4$ | $1^{10}$ |
| 24.576.17-24.bso.1.9 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{10}$ |
| 24.576.17-24.bsp.1.5 | $24$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
| 48.576.15-48.bp.1.33 | $48$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 48.576.15-48.bp.2.33 | $48$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 48.576.15-48.ca.1.30 | $48$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 48.576.15-48.ca.2.30 | $48$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 48.576.15-48.cs.1.29 | $48$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 48.576.15-48.cs.2.29 | $48$ | $2$ | $2$ | $15$ | $0$ | $2^{4}$ |
| 48.576.17-48.bq.1.30 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
| 48.576.17-48.bq.2.30 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
| 48.576.17-48.bt.1.33 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
| 48.576.17-48.bt.2.33 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
| 48.576.17-48.bu.1.29 | $48$ | $2$ | $2$ | $17$ | $4$ | $1^{10}$ |
| 48.576.17-48.bu.2.29 | $48$ | $2$ | $2$ | $17$ | $4$ | $1^{10}$ |
| 48.576.17-48.bv.1.30 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
| 48.576.17-48.bv.2.30 | $48$ | $2$ | $2$ | $17$ | $2$ | $1^{10}$ |
| 48.576.17-48.bw.1.33 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{10}$ |
| 48.576.17-48.bw.2.33 | $48$ | $2$ | $2$ | $17$ | $3$ | $1^{10}$ |
| 48.576.17-48.bx.1.29 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
| 48.576.17-48.bx.2.29 | $48$ | $2$ | $2$ | $17$ | $1$ | $1^{10}$ |
| 48.576.19-48.hu.1.30 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{2}\cdot4^{2}$ |
| 48.576.19-48.hu.2.30 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{2}\cdot4^{2}$ |
| 48.576.19-48.nm.1.29 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{2}\cdot4^{2}$ |
| 48.576.19-48.nm.2.29 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{2}\cdot4^{2}$ |
| 48.576.19-48.pl.1.33 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{2}\cdot4^{2}$ |
| 48.576.19-48.pl.2.33 | $48$ | $2$ | $2$ | $19$ | $0$ | $2^{2}\cdot4^{2}$ |