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Properties

Label 48.48.1.l.1
Level $48$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $48$ $\SL_2$-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: $2 \le \gamma \le 4$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.48.1.197

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}5&45\\22&29\end{bmatrix}$, $\begin{bmatrix}13&6\\36&43\end{bmatrix}$, $\begin{bmatrix}17&46\\44&41\end{bmatrix}$, $\begin{bmatrix}21&35\\34&5\end{bmatrix}$, $\begin{bmatrix}47&40\\0&47\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.96.1-48.l.1.1, 48.96.1-48.l.1.2, 48.96.1-48.l.1.3, 48.96.1-48.l.1.4, 48.96.1-48.l.1.5, 48.96.1-48.l.1.6, 48.96.1-48.l.1.7, 48.96.1-48.l.1.8, 48.96.1-48.l.1.9, 48.96.1-48.l.1.10, 48.96.1-48.l.1.11, 48.96.1-48.l.1.12, 48.96.1-48.l.1.13, 48.96.1-48.l.1.14, 48.96.1-48.l.1.15, 48.96.1-48.l.1.16, 240.96.1-48.l.1.1, 240.96.1-48.l.1.2, 240.96.1-48.l.1.3, 240.96.1-48.l.1.4, 240.96.1-48.l.1.5, 240.96.1-48.l.1.6, 240.96.1-48.l.1.7, 240.96.1-48.l.1.8, 240.96.1-48.l.1.9, 240.96.1-48.l.1.10, 240.96.1-48.l.1.11, 240.96.1-48.l.1.12, 240.96.1-48.l.1.13, 240.96.1-48.l.1.14, 240.96.1-48.l.1.15, 240.96.1-48.l.1.16
Cyclic 48-isogeny field degree: $32$
Cyclic 48-torsion field degree: $512$
Full 48-torsion field degree: $24576$

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 $ $=$ $ 48 x^{2} + 24 x y + 3 y^{2} - z^{2} + z w $
$=$ $48 x y - z^{2} + 2 z w + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 4 x^{3} z - 6 x^{2} y^{2} + 2 x^{2} z^{2} + 4 x z^{3} + 9 y^{4} - 6 y^{2} z^{2} + z^{4} $
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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:

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle 2y$
$\displaystyle Z$ $=$ $\displaystyle w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -2^6\,\frac{5400y^{2}z^{10}+53568y^{2}z^{9}w+113400y^{2}z^{8}w^{2}+107136y^{2}z^{7}w^{3}-32400y^{2}z^{6}w^{4}-32400y^{2}z^{4}w^{6}-107136y^{2}z^{3}w^{7}+113400y^{2}z^{2}w^{8}-53568y^{2}zw^{9}+5400y^{2}w^{10}-325z^{12}-804z^{11}w+5490z^{10}w^{2}+5588z^{9}w^{3}-4875z^{8}w^{4}-25608z^{7}w^{5}-21380z^{6}w^{6}+25608z^{5}w^{7}-4875z^{4}w^{8}-5588z^{3}w^{9}+5490z^{2}w^{10}+804zw^{11}-325w^{12}}{(z^{2}-2zw-w^{2})^{4}(24y^{2}z^{2}+24y^{2}w^{2}-3z^{4}-4z^{3}w-6z^{2}w^{2}+4zw^{3}-3w^{4})}$

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
8.24.1.l.1 $8$ $2$ $2$ $1$ $0$ 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
48.96.5.dx.1 $48$ $2$ $2$ $5$ $2$ $1^{4}$
48.96.5.dz.1 $48$ $2$ $2$ $5$ $0$ $1^{4}$
48.96.5.em.1 $48$ $2$ $2$ $5$ $0$ $1^{4}$
48.96.5.ew.1 $48$ $2$ $2$ $5$ $2$ $1^{4}$
48.96.5.fi.1 $48$ $2$ $2$ $5$ $1$ $1^{4}$
48.96.5.fs.1 $48$ $2$ $2$ $5$ $3$ $1^{4}$
48.96.5.fy.1 $48$ $2$ $2$ $5$ $3$ $1^{4}$
48.96.5.ga.1 $48$ $2$ $2$ $5$ $1$ $1^{4}$
48.144.9.bu.1 $48$ $3$ $3$ $9$ $3$ $1^{4}\cdot2^{2}$
48.192.9.pc.1 $48$ $4$ $4$ $9$ $5$ $1^{8}$
240.96.5.mq.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.ms.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.nq.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.nw.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.ow.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.pc.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.ps.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.pu.1 $240$ $2$ $2$ $5$ $?$ not computed
240.240.17.x.1 $240$ $5$ $5$ $17$ $?$ not computed
240.288.17.cdl.1 $240$ $6$ $6$ $17$ $?$ not computed