Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8G1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.1.933 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&14\\20&17\end{bmatrix}$, $\begin{bmatrix}11&2\\20&3\end{bmatrix}$, $\begin{bmatrix}13&4\\0&19\end{bmatrix}$, $\begin{bmatrix}19&18\\0&17\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2\times D_4\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.48.1.x.2 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 6 x^{2} - 6 x y - w^{2} $ |
| $=$ | $3 x y - 7 y^{2} + 4 y z - 4 z^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 28 x^{4} - 22 x^{3} z - 6 x^{2} y^{2} + 15 x^{2} z^{2} - 4 x z^{3} + z^{4} $ |
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 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{3^2\cdot7^4}\cdot\frac{6522787571466240xz^{11}-2888151357579264xz^{9}w^{2}+361858352583168xz^{7}w^{4}-51306658357248xz^{5}w^{6}+22752225182232xz^{3}w^{8}-16001617009213440y^{2}z^{10}+6246294479695872y^{2}z^{8}w^{2}-1416230802233856y^{2}z^{6}w^{4}-20644560941760y^{2}z^{4}w^{6}+31478690165328y^{2}z^{2}w^{8}-2659434619443y^{2}w^{10}+9478829437747200yz^{11}-5644878155845632yz^{9}w^{2}+1253138962295808yz^{7}w^{4}+128939829810624yz^{5}w^{6}-63697864108032yz^{3}w^{8}+4098083571645yzw^{10}-9253884417933312z^{12}+3785499048296448z^{10}w^{2}-908116905443328z^{8}w^{4}-22536520118784z^{6}w^{6}+35055598201344z^{4}w^{8}-6942111719424z^{2}w^{10}+144627327488w^{12}}{w^{4}(15040512xz^{7}+688128xz^{5}w^{2}+43512xz^{3}w^{4}-33128448y^{2}z^{6}-5562368y^{2}z^{4}w^{2}+81536y^{2}z^{2}w^{4}+1029y^{2}w^{6}+18087936yz^{7}+7380992yz^{5}w^{2}+51968yz^{3}w^{4}-5439yzw^{6}-33128448z^{8}-3928064z^{6}w^{2}-68992z^{4}w^{4})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.48.1.x.2 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2z$ |
Equation of the image curve:
| $0$ | $=$ | $ 28X^{4}-6X^{2}Y^{2}-22X^{3}Z+15X^{2}Z^{2}-4XZ^{3}+Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.1-8.d.1.3 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.0-24.h.1.2 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.h.1.9 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.i.2.2 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.i.2.17 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.1-8.d.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.192.1-24.e.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.p.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.ba.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bg.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bi.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bo.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bq.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.br.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.288.9-24.ed.2.11 | $24$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.384.9-24.cj.1.4 | $24$ | $4$ | $4$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 120.192.1-120.dm.2.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.do.2.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.dw.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ec.1.3 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.em.1.5 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.es.1.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.fa.2.9 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.fc.2.1 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.480.17-120.bv.1.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 168.192.1-168.dm.1.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.do.1.6 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.dw.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ec.1.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.em.1.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.es.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.fa.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.fc.1.2 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.dm.1.2 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.do.1.6 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.dw.1.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.ec.1.2 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.em.1.2 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.es.1.9 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.fa.1.2 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.fc.1.2 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.dm.1.2 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.do.1.6 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.dw.1.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.ec.1.2 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.em.1.2 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.es.1.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.fa.1.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.fc.1.2 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |