Invariants
Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
Index: | $48$ | $\PSL_2$-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 | $4^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | $0$ | ||||||
$\Q$-gonality: | $2 \le \gamma \le 4$ | ||||||
$\overline{\Q}$-gonality: | $2$ | ||||||
Rational cusps: | $0$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 8F1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.1.444 |
Level structure
$\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}3&4\\2&9\end{bmatrix}$, $\begin{bmatrix}13&6\\8&17\end{bmatrix}$, $\begin{bmatrix}19&3\\12&5\end{bmatrix}$ |
Contains $-I$: | yes |
Quadratic refinements: | 48.96.1-24.hd.1.1, 48.96.1-24.hd.1.2, 48.96.1-24.hd.1.3, 48.96.1-24.hd.1.4, 240.96.1-24.hd.1.1, 240.96.1-24.hd.1.2, 240.96.1-24.hd.1.3, 240.96.1-24.hd.1.4 |
Cyclic 24-isogeny field degree: | $16$ |
Cyclic 24-torsion field degree: | $128$ |
Full 24-torsion field degree: | $1536$ |
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 $ | $=$ | $ 12 x^{2} + y^{2} + y z + z^{2} $ |
$=$ | $5 y^{2} + 8 y z + 8 z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 2 x^{2} y^{2} + 6 x^{2} z^{2} + 25 y^{4} - 60 y^{2} z^{2} + 36 z^{4} $ |
Rational points
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle 2x$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle 2^8\cdot3^3\,\frac{2173878yz^{11}-1360800yz^{9}w^{2}-6671700yz^{7}w^{4}-2247000yz^{5}w^{6}-106250yz^{3}w^{8}-75000yzw^{10}+930933z^{12}-7714278z^{10}w^{2}-7424595z^{8}w^{4}+345100z^{6}w^{6}+1191875z^{4}w^{8}+131250z^{2}w^{10}-3125w^{12}}{8695512yz^{11}+19415700yz^{9}w^{2}+2567700yz^{7}w^{4}-4698000yz^{5}w^{6}-1800000yz^{3}w^{8}-300000yzw^{10}+3723732z^{12}+27535788z^{10}w^{2}+25777845z^{8}w^{4}+8199900z^{6}w^{6}+1417500z^{4}w^{8}+150000z^{2}w^{10}+50000w^{12}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
8.24.1.z.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.24.0.cr.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.cu.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.dp.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.0.ef.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
24.24.1.z.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
24.24.1.bh.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 |
---|---|---|---|---|---|---|
24.144.9.bqb.1 | $24$ | $3$ | $3$ | $9$ | $1$ | $1^{8}$ |
24.192.9.nx.1 | $24$ | $4$ | $4$ | $9$ | $2$ | $1^{8}$ |
120.240.17.zt.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
120.288.17.zbx.1 | $120$ | $6$ | $6$ | $17$ | $?$ | not computed |