Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
| 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: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.48.1.449 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}9&23\\16&7\end{bmatrix}$, $\begin{bmatrix}21&10\\4&1\end{bmatrix}$, $\begin{bmatrix}21&14\\10&11\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 24-isogeny field degree: | $16$ |
| Cyclic 24-torsion field degree: | $128$ |
| Full 24-torsion field degree: | $1536$ |
Jacobian
| Conductor: | $2^{5}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 32.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 6 x^{2} - 3 y z + w^{2} $ |
| $=$ | $6 x^{2} - y^{2} + 7 y z - z^{2} - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 20 x^{2} y^{2} - 12 x^{2} z^{2} + 4 y^{4} + 12 y^{2} z^{2} + 9 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 between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}w$ |
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^8\cdot3^3\,\frac{z^{3}(70005870yz^{8}-10459206yz^{6}w^{2}+482976yz^{4}w^{4}-7056yz^{2}w^{6}+16yw^{8}-7072029z^{9}-13233294z^{7}w^{2}+1788480z^{5}w^{4}-65800z^{3}w^{6}+592zw^{8})}{280023480yz^{11}-70124940yz^{9}w^{2}+6501060yz^{7}w^{4}-269892yz^{5}w^{6}+4716yz^{3}w^{8}-24yzw^{10}-28288116z^{12}-50075496z^{10}w^{2}+12466629z^{8}w^{4}-1051164z^{6}w^{6}+36810z^{4}w^{8}-468z^{2}w^{10}+w^{12}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.24.1.be.1 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.0.cp.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0.cv.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0.du.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.0.ek.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.24.1.bb.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.24.1.bm.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.bqk.1 | $24$ | $3$ | $3$ | $9$ | $4$ | $1^{8}$ |
| 24.192.9.og.1 | $24$ | $4$ | $4$ | $9$ | $0$ | $1^{8}$ |
| 48.96.3.lj.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3.ll.1 | $48$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 48.96.3.nv.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.nw.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.nx.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.ny.1 | $48$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 48.96.3.ol.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.3.on.1 | $48$ | $2$ | $2$ | $3$ | $2$ | $1^{2}$ |
| 48.96.5.mn.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 48.96.5.mo.1 | $48$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 120.240.17.bac.1 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 120.288.17.zcg.1 | $120$ | $6$ | $6$ | $17$ | $?$ | not computed |
| 240.96.3.bnh.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.bnj.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.bnx.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.bny.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.bnz.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.boa.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.bon.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.3.bop.1 | $240$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 240.96.5.blh.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 240.96.5.bli.1 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |