Invariants
| Level: | $48$ | $\SL_2$-level: | $48$ | Newform level: | $2304$ | ||
| Index: | $144$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $7 = 1 + \frac{ 144 }{12} - \frac{ 16 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $24^{2}\cdot48^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $16$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $7$ | ||||||
| $\Q$-gonality: | $4$ | ||||||
| $\overline{\Q}$-gonality: | $4$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | yes $\quad(D =$ $-28$) |
Other labels
| Cummins and Pauli (CP) label: | 48T7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.144.7.99 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}5&19\\32&23\end{bmatrix}$, $\begin{bmatrix}19&14\\38&41\end{bmatrix}$, $\begin{bmatrix}21&8\\2&15\end{bmatrix}$, $\begin{bmatrix}23&16\\10&25\end{bmatrix}$, $\begin{bmatrix}25&12\\36&37\end{bmatrix}$ |
| 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: | $2^{54}\cdot3^{12}$ |
| Simple: | no |
| Squarefree: | yes |
| Decomposition: | $1^{7}$ |
| Newforms: | 256.2.a.b, 576.2.a.c, 2304.2.a.a, 2304.2.a.c, 2304.2.a.g, 2304.2.a.j, 2304.2.a.m |
Models
Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ y t + z w $ |
| $=$ | $x t + y w$ | |
| $=$ | $w^{2} - t^{2} + t u$ | |
| $=$ | $x t - z t + z u$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{10} + 6 x^{8} z^{2} + x^{6} y^{4} + 12 x^{6} z^{4} - 9 x^{4} y^{4} z^{2} + 36 x^{4} z^{6} + \cdots + 54 z^{10} $ |
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:
| $\displaystyle X$ | $=$ | $\displaystyle z$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{3}v$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{3}t$ |
Maps to other modular curves
$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^6\cdot3^3\,\frac{597168z^{2}u^{10}+800928z^{2}u^{6}v^{4}+264384z^{2}u^{2}v^{8}+36792wtu^{8}v^{2}+71856wtu^{4}v^{6}+8991wtv^{10}+47744wu^{9}v^{2}+87456wu^{5}v^{6}+40797wuv^{10}+6344t^{2}u^{10}-4536t^{2}u^{6}v^{4}-32940t^{2}u^{2}v^{8}+66352tu^{11}+45240tu^{7}v^{4}+21204tu^{3}v^{8}+1944u^{12}+18608u^{8}v^{4}+12696u^{4}v^{8}-9v^{12}}{2304z^{2}u^{10}+62208z^{2}u^{6}v^{4}+64152z^{2}u^{2}v^{8}+2880wtu^{8}v^{2}+12672wtu^{4}v^{6}-4617wtv^{10}+1088wu^{9}v^{2}+14016wu^{5}v^{6}+12285wuv^{10}+512t^{2}u^{10}+14688t^{2}u^{6}v^{4}+9180t^{2}u^{2}v^{8}+256tu^{11}+8736tu^{7}v^{4}+27252tu^{3}v^{8}-832u^{8}v^{4}-5664u^{4}v^{8}-9v^{12}}$ |
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 |
|---|---|---|---|---|---|---|
| 24.72.1.fb.1 | $24$ | $2$ | $2$ | $1$ | $1$ | $1^{6}$ |
| 48.72.3.bl.1 | $48$ | $2$ | $2$ | $3$ | $3$ | $1^{4}$ |
| 48.72.3.bn.1 | $48$ | $2$ | $2$ | $3$ | $3$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 48.288.13.qg.1 | $48$ | $2$ | $2$ | $13$ | $7$ | $1^{4}\cdot2$ |
| 48.288.13.qi.1 | $48$ | $2$ | $2$ | $13$ | $8$ | $1^{4}\cdot2$ |
| 48.288.13.rp.1 | $48$ | $2$ | $2$ | $13$ | $7$ | $1^{4}\cdot2$ |
| 48.288.13.rr.1 | $48$ | $2$ | $2$ | $13$ | $8$ | $1^{4}\cdot2$ |
| 48.288.13.bcc.1 | $48$ | $2$ | $2$ | $13$ | $10$ | $1^{4}\cdot2$ |
| 48.288.13.bce.1 | $48$ | $2$ | $2$ | $13$ | $11$ | $1^{4}\cdot2$ |
| 48.288.13.bhh.1 | $48$ | $2$ | $2$ | $13$ | $9$ | $1^{4}\cdot2$ |
| 48.288.13.bhj.1 | $48$ | $2$ | $2$ | $13$ | $10$ | $1^{4}\cdot2$ |
| 48.288.21.dt.2 | $48$ | $2$ | $2$ | $21$ | $7$ | $1^{14}$ |
| 48.288.21.bdc.1 | $48$ | $2$ | $2$ | $21$ | $17$ | $1^{14}$ |
| 48.288.21.byv.2 | $48$ | $2$ | $2$ | $21$ | $7$ | $1^{14}$ |
| 48.288.21.bzl.1 | $48$ | $2$ | $2$ | $21$ | $16$ | $1^{14}$ |
| 48.288.21.dvp.1 | $48$ | $2$ | $2$ | $21$ | $8$ | $1^{4}\cdot2^{3}\cdot4$ |
| 48.288.21.dvr.1 | $48$ | $2$ | $2$ | $21$ | $7$ | $1^{4}\cdot2^{3}\cdot4$ |
| 48.288.21.eas.1 | $48$ | $2$ | $2$ | $21$ | $9$ | $1^{4}\cdot2^{3}\cdot4$ |
| 48.288.21.eau.1 | $48$ | $2$ | $2$ | $21$ | $8$ | $1^{4}\cdot2^{3}\cdot4$ |
| 48.288.21.gxl.1 | $48$ | $2$ | $2$ | $21$ | $12$ | $1^{14}$ |
| 48.288.21.gxn.2 | $48$ | $2$ | $2$ | $21$ | $12$ | $1^{14}$ |
| 48.288.21.gyj.2 | $48$ | $2$ | $2$ | $21$ | $12$ | $1^{14}$ |
| 48.288.21.gyn.2 | $48$ | $2$ | $2$ | $21$ | $12$ | $1^{14}$ |
| 48.288.21.hln.1 | $48$ | $2$ | $2$ | $21$ | $11$ | $1^{4}\cdot2^{3}\cdot4$ |
| 48.288.21.hlp.1 | $48$ | $2$ | $2$ | $21$ | $11$ | $1^{4}\cdot2^{3}\cdot4$ |
| 48.288.21.hmu.1 | $48$ | $2$ | $2$ | $21$ | $12$ | $1^{4}\cdot2^{3}\cdot4$ |
| 48.288.21.hmw.1 | $48$ | $2$ | $2$ | $21$ | $12$ | $1^{4}\cdot2^{3}\cdot4$ |
| 240.288.13.foi.1 | $240$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.288.13.fok.1 | $240$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.288.13.fqf.1 | $240$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.288.13.fqh.1 | $240$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.288.13.grw.1 | $240$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.288.13.gry.1 | $240$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.288.13.guz.1 | $240$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.288.13.gvb.1 | $240$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.288.21.bsxd.2 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.bsxh.2 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.bsyj.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.bsyn.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.btwb.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.btwd.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.btzc.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.btze.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.buyz.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.buzd.2 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.bvaf.2 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.bvaj.2 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.bvyn.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.bvyp.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.bwai.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.bwak.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |