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Properties

Label 48.48.1.bp.1
Level $48$
Index $48$
Genus $1$
Analytic rank $1$
Cusps $8$
$\Q$-cusps $2$

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Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $576$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $2$ are rational) Cusp widths $2^{4}\cdot4^{2}\cdot16^{2}$ Cusp orbits $1^{2}\cdot2^{3}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 16G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.48.1.287

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}11&42\\44&41\end{bmatrix}$, $\begin{bmatrix}29&8\\36&43\end{bmatrix}$, $\begin{bmatrix}29&32\\20&39\end{bmatrix}$, $\begin{bmatrix}31&11\\36&47\end{bmatrix}$, $\begin{bmatrix}47&17\\4&15\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.96.1-48.bp.1.1, 48.96.1-48.bp.1.2, 48.96.1-48.bp.1.3, 48.96.1-48.bp.1.4, 48.96.1-48.bp.1.5, 48.96.1-48.bp.1.6, 48.96.1-48.bp.1.7, 48.96.1-48.bp.1.8, 48.96.1-48.bp.1.9, 48.96.1-48.bp.1.10, 48.96.1-48.bp.1.11, 48.96.1-48.bp.1.12, 48.96.1-48.bp.1.13, 48.96.1-48.bp.1.14, 48.96.1-48.bp.1.15, 48.96.1-48.bp.1.16, 240.96.1-48.bp.1.1, 240.96.1-48.bp.1.2, 240.96.1-48.bp.1.3, 240.96.1-48.bp.1.4, 240.96.1-48.bp.1.5, 240.96.1-48.bp.1.6, 240.96.1-48.bp.1.7, 240.96.1-48.bp.1.8, 240.96.1-48.bp.1.9, 240.96.1-48.bp.1.10, 240.96.1-48.bp.1.11, 240.96.1-48.bp.1.12, 240.96.1-48.bp.1.13, 240.96.1-48.bp.1.14, 240.96.1-48.bp.1.15, 240.96.1-48.bp.1.16
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $24576$

Jacobian

Conductor: $2^{6}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 576.2.a.c

Models

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ x^{3} - 396x - 3024 $
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Rational points

This modular curve has infinitely many rational points, including 1 stored non-cuspidal point.

Maps to other modular curves

$j$-invariant map of degree 48 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2\cdot3}\cdot\frac{4176x^{2}y^{14}-56713149216x^{2}y^{12}z^{2}-887327030744064x^{2}y^{10}z^{4}+54284024152847563776x^{2}y^{8}z^{6}+24604992849854578950144x^{2}y^{6}z^{8}-18094983667622272994205302784x^{2}y^{4}z^{10}-166825072115400751987968876478464x^{2}y^{2}z^{12}-421596930836809960564255254978232320x^{2}z^{14}-5843376xy^{14}z-672089571072xy^{12}z^{3}+1395590517432576xy^{10}z^{5}+1581183453247639031808xy^{8}z^{7}-9292237424814369795342336xy^{6}z^{9}-538828321748326405461425258496xy^{4}z^{11}-4187262207135020604110150594199552xy^{2}z^{13}-9684318754352320554987748010250731520xz^{15}-y^{16}+2780085888y^{14}z^{2}-53864982229248y^{12}z^{4}+1366084361822294016y^{10}z^{6}+18004944846034226331648y^{8}z^{8}-538760522078390730768777216y^{6}z^{10}-8340260473011422545944664080384y^{4}z^{12}-38280102371311369824029071066005504y^{2}z^{14}-55501867011727212343338600744464941056z^{16}}{y^{2}(x^{2}y^{12}+3861216x^{2}y^{10}z^{2}+691565185152x^{2}y^{8}z^{4}+27094448757030912x^{2}y^{6}z^{6}+338068857005747712000x^{2}y^{4}z^{8}+1274480657251757342982144x^{2}y^{2}z^{10}+101559956668416x^{2}z^{12}+288xy^{12}z+276097248xy^{10}z^{3}+29276854617600xy^{8}z^{5}+854676761417607168xy^{6}z^{7}+8839384884280647843840xy^{4}z^{9}+29275537909624439332995072xy^{2}z^{11}-1218719480020992xz^{13}+40896y^{12}z^{2}+14614774272y^{10}z^{4}+857714720419584y^{8}z^{6}+15143752147096977408y^{6}z^{8}+93835088332229428051968y^{4}z^{10}+167781240271477187838738432y^{2}z^{12}-25593109080440832z^{14})}$

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
16.24.0.e.2 $16$ $2$ $2$ $0$ $0$ full Jacobian
24.24.0.bz.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.24.1.a.1 $48$ $2$ $2$ $1$ $1$ 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
48.96.1.o.2 $48$ $2$ $2$ $1$ $1$ dimension zero
48.96.1.u.2 $48$ $2$ $2$ $1$ $1$ dimension zero
48.96.1.bf.1 $48$ $2$ $2$ $1$ $1$ dimension zero
48.96.1.bw.1 $48$ $2$ $2$ $1$ $1$ dimension zero
48.96.1.dp.1 $48$ $2$ $2$ $1$ $1$ dimension zero
48.96.1.du.2 $48$ $2$ $2$ $1$ $1$ dimension zero
48.96.1.eg.1 $48$ $2$ $2$ $1$ $1$ dimension zero
48.96.1.ej.1 $48$ $2$ $2$ $1$ $1$ dimension zero
48.144.9.ir.2 $48$ $3$ $3$ $9$ $1$ $1^{4}\cdot2^{2}$
48.192.9.bfi.1 $48$ $4$ $4$ $9$ $2$ $1^{4}\cdot2^{2}$
240.96.1.nw.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.oe.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.pc.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.pk.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.su.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.tc.2 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ua.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.96.1.ui.1 $240$ $2$ $2$ $1$ $?$ dimension zero
240.240.17.el.2 $240$ $5$ $5$ $17$ $?$ not computed
240.288.17.ijt.1 $240$ $6$ $6$ $17$ $?$ not computed