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: | none |
Other labels
| Cummins and Pauli (CP) label: | 48T7 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 48.144.7.97 |
Level structure
| $\GL_2(\Z/48\Z)$-generators: | $\begin{bmatrix}13&45\\0&43\end{bmatrix}$, $\begin{bmatrix}29&0\\12&17\end{bmatrix}$, $\begin{bmatrix}29&2\\40&41\end{bmatrix}$, $\begin{bmatrix}35&23\\38&13\end{bmatrix}$, $\begin{bmatrix}47&33\\36&29\end{bmatrix}$ |
| 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 $ | $=$ | $ x^{2} + z w $ |
| $=$ | $x^{2} - x y + z t$ | |
| $=$ | $z t + w^{2} - w t - w u$ | |
| $=$ | $x w - x t - y w$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 49 x^{8} - 28 x^{7} y + 88 x^{6} y^{2} - 136 x^{5} y^{3} + 124 x^{4} y^{4} - 4 x^{4} z^{4} + \cdots + 8 y^{4} 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 canonical model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle v$ |
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 3^3\,\frac{7735687690759761744xtu^{8}v^{2}+584989867550425656xtu^{4}v^{6}+2120518785511665xtv^{10}-10030829331625918272xu^{9}v^{2}-465589582859582208xu^{5}v^{6}-2448325234526048xuv^{10}+11492727565487438160ytu^{8}v^{2}+279706063718504280ytu^{4}v^{6}-389302214588595ytv^{10}+4459536531189279504yu^{9}v^{2}+762075301955948280yu^{5}v^{6}+4869292102417249yuv^{10}-43038404650366873344ztu^{10}-1589775032658398528ztu^{6}v^{4}-15101327628612752ztu^{2}v^{8}-19136981009275683840zu^{11}+2172860472944145600zu^{7}v^{4}+49561679444013440zu^{3}v^{8}+57025552828962069504wtu^{10}+4867692494469016896wtu^{6}v^{4}+12098804211202464wtu^{2}v^{8}+91422338777397462528wu^{11}+12874659224463399296wu^{7}v^{4}+146790635766515440wu^{3}v^{8}-10283105531331581184t^{3}u^{9}+321978529835210880t^{3}u^{5}v^{4}-949970348456208t^{3}uv^{8}+38147466490572646656t^{2}u^{10}+4869390836869479296t^{2}u^{6}v^{4}+52502539063173680t^{2}u^{2}v^{8}-53274872286824815872tu^{11}-4394242318063155456tu^{7}v^{4}+12791552576893344tu^{3}v^{8}+14009251482431301888u^{12}+3078590168744506416u^{8}v^{4}+106114347981569576u^{4}v^{8}+123278277356971v^{12}}{422045576518919760xtu^{8}v^{2}-25823802636028980xtu^{4}v^{6}+36791938665405xtv^{10}+20457270365685120xu^{9}v^{2}+6509284295176512xu^{5}v^{6}-72064439509536xuv^{10}-79697438823690864ytu^{8}v^{2}+7419841702820316ytu^{4}v^{6}-8820030275991ytv^{10}+294616387556661456yu^{9}v^{2}-28301367299933556yu^{5}v^{6}+127345368747789yuv^{10}-150778127157926400ztu^{10}+40723785962455680ztu^{6}v^{4}-67936778251392ztu^{2}v^{8}-14196168560559744zu^{7}v^{4}+588043771748928zu^{3}v^{8}+510877502956212480wtu^{10}-137113190727459264wtu^{6}v^{4}+1131284894751216wtu^{2}v^{8}+777726543046210560wu^{11}-261316918909168128wu^{7}v^{4}+3497499933185568wu^{3}v^{8}+24067764553003776t^{3}u^{9}+1827952628275584t^{3}u^{5}v^{4}+2210079434256t^{3}uv^{8}+290916804643001856t^{2}u^{10}-102810399474383424t^{2}u^{6}v^{4}+1143658876005840t^{2}u^{2}v^{8}-486809738403208704tu^{11}+191987631850239936tu^{7}v^{4}-3447995048724672tu^{3}v^{8}+46865082794094576u^{8}v^{4}-2096122061546172u^{4}v^{8}+2037536385071v^{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.2 | $48$ | $2$ | $2$ | $3$ | $3$ | $1^{4}$ |
| 48.72.3.bn.2 | $48$ | $2$ | $2$ | $3$ | $3$ | $1^{4}$ |
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.qh.1 | $48$ | $2$ | $2$ | $13$ | $7$ | $1^{4}\cdot2$ |
| 48.288.13.qj.1 | $48$ | $2$ | $2$ | $13$ | $7$ | $1^{4}\cdot2$ |
| 48.288.13.ro.1 | $48$ | $2$ | $2$ | $13$ | $8$ | $1^{4}\cdot2$ |
| 48.288.13.rq.1 | $48$ | $2$ | $2$ | $13$ | $8$ | $1^{4}\cdot2$ |
| 48.288.13.bcd.1 | $48$ | $2$ | $2$ | $13$ | $10$ | $1^{4}\cdot2$ |
| 48.288.13.bcf.1 | $48$ | $2$ | $2$ | $13$ | $9$ | $1^{4}\cdot2$ |
| 48.288.13.bhg.1 | $48$ | $2$ | $2$ | $13$ | $11$ | $1^{4}\cdot2$ |
| 48.288.13.bhi.1 | $48$ | $2$ | $2$ | $13$ | $10$ | $1^{4}\cdot2$ |
| 48.288.21.dt.1 | $48$ | $2$ | $2$ | $21$ | $7$ | $1^{14}$ |
| 48.288.21.bdc.1 | $48$ | $2$ | $2$ | $21$ | $17$ | $1^{14}$ |
| 48.288.21.byv.1 | $48$ | $2$ | $2$ | $21$ | $7$ | $1^{14}$ |
| 48.288.21.bzl.1 | $48$ | $2$ | $2$ | $21$ | $16$ | $1^{14}$ |
| 48.288.21.dvo.1 | $48$ | $2$ | $2$ | $21$ | $8$ | $1^{4}\cdot2^{3}\cdot4$ |
| 48.288.21.dvq.1 | $48$ | $2$ | $2$ | $21$ | $9$ | $1^{4}\cdot2^{3}\cdot4$ |
| 48.288.21.eat.1 | $48$ | $2$ | $2$ | $21$ | $7$ | $1^{4}\cdot2^{3}\cdot4$ |
| 48.288.21.eav.1 | $48$ | $2$ | $2$ | $21$ | $8$ | $1^{4}\cdot2^{3}\cdot4$ |
| 48.288.21.gxl.2 | $48$ | $2$ | $2$ | $21$ | $12$ | $1^{14}$ |
| 48.288.21.gxn.1 | $48$ | $2$ | $2$ | $21$ | $12$ | $1^{14}$ |
| 48.288.21.gyj.1 | $48$ | $2$ | $2$ | $21$ | $12$ | $1^{14}$ |
| 48.288.21.gyn.1 | $48$ | $2$ | $2$ | $21$ | $12$ | $1^{14}$ |
| 48.288.21.hlm.1 | $48$ | $2$ | $2$ | $21$ | $11$ | $1^{4}\cdot2^{3}\cdot4$ |
| 48.288.21.hlo.1 | $48$ | $2$ | $2$ | $21$ | $12$ | $1^{4}\cdot2^{3}\cdot4$ |
| 48.288.21.hmv.1 | $48$ | $2$ | $2$ | $21$ | $11$ | $1^{4}\cdot2^{3}\cdot4$ |
| 48.288.21.hmx.1 | $48$ | $2$ | $2$ | $21$ | $12$ | $1^{4}\cdot2^{3}\cdot4$ |
| 240.288.13.foj.1 | $240$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.288.13.fol.1 | $240$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.288.13.fqe.1 | $240$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.288.13.fqg.1 | $240$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.288.13.grx.1 | $240$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.288.13.grz.1 | $240$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.288.13.guy.1 | $240$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.288.13.gva.1 | $240$ | $2$ | $2$ | $13$ | $?$ | not computed |
| 240.288.21.bsxd.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.bsxh.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.bsyj.2 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.bsyn.2 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.btwa.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.btwc.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.btzd.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.btzf.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.buyz.2 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.buzd.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.bvaf.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.bvaj.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.bvym.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.bvyo.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.bwaj.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |
| 240.288.21.bwal.1 | $240$ | $2$ | $2$ | $21$ | $?$ | not computed |