Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $144$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $5 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
| Cusps: | $16$ (of which $4$ are rational) | Cusp widths | $6^{8}\cdot12^{8}$ | Cusp orbits | $1^{4}\cdot2^{4}\cdot4$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $4$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12B5 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.288.5.861 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}7&18\\12&23\end{bmatrix}$, $\begin{bmatrix}13&18\\12&17\end{bmatrix}$, $\begin{bmatrix}17&9\\0&23\end{bmatrix}$, $\begin{bmatrix}19&0\\12&19\end{bmatrix}$, $\begin{bmatrix}19&6\\0&7\end{bmatrix}$, $\begin{bmatrix}19&9\\0&1\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4^2:C_2^4$ |
| Contains $-I$: | no $\quad$ (see 12.144.5.n.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: | $2^{18}\cdot3^{8}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{5}$ |
| Newforms: | 36.2.a.a, 48.2.a.a$^{2}$, 144.2.a.a, 144.2.a.b |
Models
Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x^{2} - y^{2} + z w - z t + w t $ |
| $=$ | $x^{2} + 2 y^{2} + z^{2} + w t$ | |
| $=$ | $x^{2} - y^{2} + z^{2} - z w + z t + w^{2} + 2 w t + t^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} z^{4} - 6 x^{3} y^{2} z^{3} - 2 x^{3} z^{5} + 9 x^{2} y^{4} z^{2} + 10 x^{2} y^{2} z^{4} + \cdots + 2 z^{8} $ |
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 |
|---|
| $(0:1/3:-1/3:-1/3:1)$, $(0:1:-1:-3:1)$, $(0:-1/3:-1/3:-1/3:1)$, $(0:-1:-1:-3:1)$ |
Maps to other modular curves
$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -3^3\,\frac{108zw^{17}-7164zw^{16}t+129888zw^{15}t^{2}-1039680zw^{14}t^{3}+4608928zw^{13}t^{4}-13193856zw^{12}t^{5}+27285120zw^{11}t^{6}-43341024zw^{10}t^{7}+54217080zw^{9}t^{8}-54217080zw^{8}t^{9}+43341024zw^{7}t^{10}-27285120zw^{6}t^{11}+13193856zw^{5}t^{12}-4608928zw^{4}t^{13}+1039680zw^{3}t^{14}-129888zw^{2}t^{15}+7164zwt^{16}-108zt^{17}-99w^{18}+5382w^{17}t-77355w^{16}t^{2}+476352w^{15}t^{3}-1609372w^{14}t^{4}+3651976w^{13}t^{5}-6425676w^{12}t^{6}+9188544w^{11}t^{7}-11146890w^{10}t^{8}+11911140w^{9}t^{9}-11146890w^{8}t^{10}+9188544w^{7}t^{11}-6425676w^{6}t^{12}+3651976w^{5}t^{13}-1609372w^{4}t^{14}+476352w^{3}t^{15}-77355w^{2}t^{16}+5382wt^{17}-99t^{18}}{t^{3}w^{3}(w-t)^{3}(90zw^{8}-3808zw^{7}t+37914zw^{6}t^{2}-136212zw^{5}t^{3}+205760zw^{4}t^{4}-136212zw^{3}t^{5}+37914zw^{2}t^{6}-3808zwt^{7}+90zt^{8}-81w^{9}+2620w^{8}t-17692w^{7}t^{2}+34563w^{6}t^{3}-20312w^{5}t^{4}+20312w^{4}t^{5}-34563w^{3}t^{6}+17692w^{2}t^{7}-2620wt^{8}+81t^{9})}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 12.144.5.n.1 :
| $\displaystyle X$ | $=$ | $\displaystyle x+w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle z$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{4}Z^{4}-6X^{3}Y^{2}Z^{3}-2X^{3}Z^{5}+9X^{2}Y^{4}Z^{2}+10X^{2}Y^{2}Z^{4}+3X^{2}Z^{6}-12XY^{4}Z^{3}-10XY^{2}Z^{5}-2XZ^{7}+9Y^{8}-24Y^{6}Z^{2}+10Y^{4}Z^{4}+12Y^{2}Z^{6}+2Z^{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 |
|---|---|---|---|---|---|---|
| $X_{\mathrm{sp}}(3)$ | $3$ | $24$ | $12$ | $0$ | $0$ | full Jacobian |
| 8.24.0-4.d.1.2 | $8$ | $12$ | $12$ | $0$ | $0$ | full Jacobian |
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.96.1-12.h.1.23 | $24$ | $3$ | $3$ | $1$ | $0$ | $1^{4}$ |
| 24.144.1-12.f.1.11 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{4}$ |
| 24.144.1-12.f.1.17 | $24$ | $2$ | $2$ | $1$ | $0$ | $1^{4}$ |
| 24.144.3-12.bf.1.9 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.144.3-12.bf.1.17 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.144.3-12.cc.1.2 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.144.3-12.cc.1.12 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.576.9-12.f.1.8 | $24$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 24.576.9-12.f.1.14 | $24$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 24.576.9-24.m.1.6 | $24$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 24.576.9-24.m.1.13 | $24$ | $2$ | $2$ | $9$ | $0$ | $2^{2}$ |
| 24.576.13-12.r.1.10 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{8}$ |
| 24.576.13-12.s.1.9 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 24.576.13-12.t.1.10 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{8}$ |
| 24.576.13-24.gm.1.1 | $24$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 24.576.13-24.go.1.6 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{8}$ |
| 24.576.13-24.hm.1.5 | $24$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 24.576.13-24.hn.1.6 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{8}$ |
| 24.576.13-24.hz.1.13 | $24$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 24.576.13-24.ia.1.13 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 24.576.13-24.ib.1.4 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 24.576.13-24.ib.1.9 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 24.576.13-24.ic.1.5 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 24.576.13-24.ic.1.10 | $24$ | $2$ | $2$ | $13$ | $0$ | $2^{4}$ |
| 24.576.13-24.id.1.9 | $24$ | $2$ | $2$ | $13$ | $3$ | $1^{8}$ |
| 24.576.13-24.is.1.7 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{8}$ |
| 24.576.13-24.it.1.6 | $24$ | $2$ | $2$ | $13$ | $4$ | $1^{8}$ |
| 24.576.13-24.ji.1.7 | $24$ | $2$ | $2$ | $13$ | $0$ | $1^{8}$ |
| 24.576.13-24.jk.1.2 | $24$ | $2$ | $2$ | $13$ | $2$ | $1^{8}$ |
| 24.576.17-24.cto.1.16 | $24$ | $2$ | $2$ | $17$ | $2$ | $1^{12}$ |
| 24.576.17-24.ctp.1.16 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{12}$ |
| 24.576.17-24.ctq.1.13 | $24$ | $2$ | $2$ | $17$ | $0$ | $4^{3}$ |
| 24.576.17-24.ctr.1.13 | $24$ | $2$ | $2$ | $17$ | $0$ | $4^{3}$ |
| 24.576.17-24.cts.1.14 | $24$ | $2$ | $2$ | $17$ | $4$ | $1^{12}$ |
| 24.576.17-24.ctt.1.14 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{12}$ |