Invariants
| Level: | $48$ | $\SL_2$-level: | $24$ | Newform level: | $2304$ | ||
| Index: | $144$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $8 = 1 + \frac{ 144 }{12} - \frac{ 8 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (none of which are rational) | Cusp widths | $24^{6}$ | Cusp orbits | $2\cdot4$ | ||
| Elliptic points: | $8$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $7$ | ||||||
| $\Q$-gonality: | $3 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $3 \le \gamma \le 6$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-4$) |
Other labels
| Cummins and Pauli (CP) label: | 24K8 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.144.8.333 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}3&16\\28&3\end{bmatrix}$, $\begin{bmatrix}7&11\\22&37\end{bmatrix}$, $\begin{bmatrix}15&5\\10&33\end{bmatrix}$, $\begin{bmatrix}17&21\\24&7\end{bmatrix}$, $\begin{bmatrix}21&31\\32&15\end{bmatrix}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 48-isogeny field degree: | $32$ |
| Cyclic 48-torsion field degree: | $512$ |
| Full 48-torsion field degree: | $8192$ |
Jacobian
| Conductor: | $2^{57}\cdot3^{16}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{4}\cdot2^{2}$ |
| Newforms: | 288.2.a.d, 576.2.a.b, 576.2.a.e, 2304.2.a.h, 2304.2.a.q, 2304.2.a.s |
Models
Canonical model in $\mathbb{P}^{ 7 }$ defined by 15 equations
| $ 0 $ | $=$ | $ x v - y t $ |
| $=$ | $y w + y r - z v$ | |
| $=$ | $x w + x r - z t$ | |
| $=$ | $x w - x r - y w + z t + z u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - 2 x^{6} y^{8} + 9 x^{6} y^{6} z^{2} - 8 x^{6} y^{4} z^{4} + 2 x^{6} y^{2} z^{6} - 36 x^{4} y^{10} + \cdots + 8 z^{14} $ |
Rational points
This modular curve has 1 rational CM point but no rational cusps or other known rational points.
Maps between models of this curve
Birational map from canonical model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x+w$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{2}z$ |
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 24.72.3.bhk.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -3z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -w$ |
| $\displaystyle Z$ | $=$ | $\displaystyle r$ |
Equation of the image curve:
| $0$ | $=$ | $ 2X^{4}+6X^{2}Y^{2}+6Y^{4}-5X^{2}Z^{2}-9Y^{2}Z^{2}+3Z^{4} $ |
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 |
|---|---|---|---|---|---|---|
| 24.72.3.bhk.1 | $24$ | $2$ | $2$ | $3$ | $2$ | $1\cdot2^{2}$ |
| 48.72.3.a.2 | $48$ | $2$ | $2$ | $3$ | $3$ | $1^{3}\cdot2$ |
| 48.72.4.bs.1 | $48$ | $2$ | $2$ | $4$ | $4$ | $1^{2}\cdot2$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.288.15.bno.1 | $48$ | $2$ | $2$ | $15$ | $10$ | $1^{5}\cdot2$ |
| 48.288.15.boe.1 | $48$ | $2$ | $2$ | $15$ | $10$ | $1^{5}\cdot2$ |
| 48.288.15.bqt.1 | $48$ | $2$ | $2$ | $15$ | $10$ | $1^{5}\cdot2$ |
| 48.288.15.brj.1 | $48$ | $2$ | $2$ | $15$ | $9$ | $1^{5}\cdot2$ |
| 48.288.15.csc.1 | $48$ | $2$ | $2$ | $15$ | $13$ | $1^{5}\cdot2$ |
| 48.288.15.csm.1 | $48$ | $2$ | $2$ | $15$ | $10$ | $1^{5}\cdot2$ |
| 48.288.15.dap.1 | $48$ | $2$ | $2$ | $15$ | $8$ | $1^{5}\cdot2$ |
| 48.288.15.daz.1 | $48$ | $2$ | $2$ | $15$ | $9$ | $1^{5}\cdot2$ |
| 48.288.19.en.1 | $48$ | $2$ | $2$ | $19$ | $14$ | $1^{7}\cdot2^{2}$ |
| 48.288.19.vm.1 | $48$ | $2$ | $2$ | $19$ | $11$ | $1^{7}\cdot2^{2}$ |
| 48.288.19.byb.2 | $48$ | $2$ | $2$ | $19$ | $9$ | $1^{7}\cdot2^{2}$ |
| 48.288.19.byc.1 | $48$ | $2$ | $2$ | $19$ | $10$ | $1^{7}\cdot2^{2}$ |
| 48.288.19.czv.1 | $48$ | $2$ | $2$ | $19$ | $9$ | $1^{9}\cdot2$ |
| 48.288.19.dad.1 | $48$ | $2$ | $2$ | $19$ | $8$ | $1^{9}\cdot2$ |
| 48.288.19.dfy.1 | $48$ | $2$ | $2$ | $19$ | $11$ | $1^{9}\cdot2$ |
| 48.288.19.dgg.1 | $48$ | $2$ | $2$ | $19$ | $15$ | $1^{9}\cdot2$ |
| 48.288.19.gdt.1 | $48$ | $2$ | $2$ | $19$ | $10$ | $1^{7}\cdot2^{2}$ |
| 48.288.19.gej.1 | $48$ | $2$ | $2$ | $19$ | $9$ | $1^{7}\cdot2^{2}$ |
| 48.288.19.ghl.1 | $48$ | $2$ | $2$ | $19$ | $8$ | $1^{7}\cdot2^{2}$ |
| 48.288.19.gib.1 | $48$ | $2$ | $2$ | $19$ | $12$ | $1^{7}\cdot2^{2}$ |
| 48.288.19.gqz.1 | $48$ | $2$ | $2$ | $19$ | $10$ | $1^{9}\cdot2$ |
| 48.288.19.grt.1 | $48$ | $2$ | $2$ | $19$ | $10$ | $1^{9}\cdot2$ |
| 48.288.19.guy.1 | $48$ | $2$ | $2$ | $19$ | $11$ | $1^{9}\cdot2$ |
| 48.288.19.gvo.1 | $48$ | $2$ | $2$ | $19$ | $12$ | $1^{9}\cdot2$ |
| 240.288.15.mom.1 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.288.15.mps.1 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.288.15.mvh.1 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.288.15.mwn.1 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.288.15.ods.1 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.288.15.oey.1 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.288.15.olt.1 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.288.15.omz.1 | $240$ | $2$ | $2$ | $15$ | $?$ | not computed |
| 240.288.19.bsnq.1 | $240$ | $2$ | $2$ | $19$ | $?$ | not computed |
| 240.288.19.bsow.1 | $240$ | $2$ | $2$ | $19$ | $?$ | not computed |
| 240.288.19.bswg.1 | $240$ | $2$ | $2$ | $19$ | $?$ | not computed |
| 240.288.19.bsxm.1 | $240$ | $2$ | $2$ | $19$ | $?$ | not computed |
| 240.288.19.btmx.1 | $240$ | $2$ | $2$ | $19$ | $?$ | not computed |
| 240.288.19.btod.1 | $240$ | $2$ | $2$ | $19$ | $?$ | not computed |
| 240.288.19.btuw.1 | $240$ | $2$ | $2$ | $19$ | $?$ | not computed |
| 240.288.19.btwc.1 | $240$ | $2$ | $2$ | $19$ | $?$ | not computed |
| 240.288.19.bvcs.1 | $240$ | $2$ | $2$ | $19$ | $?$ | not computed |
| 240.288.19.bvdy.1 | $240$ | $2$ | $2$ | $19$ | $?$ | not computed |
| 240.288.19.bvli.1 | $240$ | $2$ | $2$ | $19$ | $?$ | not computed |
| 240.288.19.bvmo.1 | $240$ | $2$ | $2$ | $19$ | $?$ | not computed |
| 240.288.19.bwcp.1 | $240$ | $2$ | $2$ | $19$ | $?$ | not computed |
| 240.288.19.bwdv.1 | $240$ | $2$ | $2$ | $19$ | $?$ | not computed |
| 240.288.19.bwji.1 | $240$ | $2$ | $2$ | $19$ | $?$ | not computed |
| 240.288.19.bwko.1 | $240$ | $2$ | $2$ | $19$ | $?$ | not computed |