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: | 24.96.1.909 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&14\\8&23\end{bmatrix}$, $\begin{bmatrix}3&8\\20&21\end{bmatrix}$, $\begin{bmatrix}15&14\\20&5\end{bmatrix}$, $\begin{bmatrix}23&22\\0&19\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_2\times D_4\times \GL(2,3)$ |
| Contains $-I$: | no $\quad$ (see 24.48.1.y.1 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 no real points, and therefore no 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.y.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2z$ |
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.1.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.0-24.i.1.4 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.i.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.2.25 | $24$ | $2$ | $2$ | $0$ | $0$ | 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.p.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.p.2.8 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.be.1.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.be.2.8 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bm.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bm.2.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.br.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.br.2.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.288.9-24.eh.1.11 | $24$ | $3$ | $3$ | $9$ | $0$ | $1^{4}\cdot2^{2}$ |
| 24.384.9-24.cl.1.4 | $24$ | $4$ | $4$ | $9$ | $1$ | $1^{4}\cdot2^{2}$ |
| 120.192.1-120.dq.1.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.dq.2.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ee.1.6 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.ee.2.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.eu.1.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.eu.2.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.fe.1.11 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.fe.2.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.480.17-120.bx.1.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 168.192.1-168.dq.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.dq.2.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ee.1.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.ee.2.8 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.eu.1.6 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.eu.2.6 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.fe.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.fe.2.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.dq.1.8 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.dq.2.8 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.ee.1.8 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.ee.2.8 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.eu.1.6 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.eu.2.10 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.fe.1.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.fe.2.4 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.dq.1.8 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.dq.2.8 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.ee.1.8 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.ee.2.8 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.eu.1.10 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.eu.2.10 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.fe.1.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.fe.2.4 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |