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Properties

Label 48.96.1.i.1
Level $48$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $288$
Index: $96$ $\PSL_2$-index:$96$
Genus: $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$
Cusps: $16$ (none of which are rational) Cusp widths $2^{8}\cdot4^{4}\cdot16^{4}$ Cusp orbits $2^{6}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 16M1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 48.96.1.63

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}9&26\\16&5\end{bmatrix}$, $\begin{bmatrix}17&2\\24&13\end{bmatrix}$, $\begin{bmatrix}33&32\\4&15\end{bmatrix}$, $\begin{bmatrix}39&28\\32&47\end{bmatrix}$, $\begin{bmatrix}43&16\\0&47\end{bmatrix}$, $\begin{bmatrix}47&32\\24&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.i.1.1, 48.192.1-48.i.1.2, 48.192.1-48.i.1.3, 48.192.1-48.i.1.4, 48.192.1-48.i.1.5, 48.192.1-48.i.1.6, 48.192.1-48.i.1.7, 48.192.1-48.i.1.8, 48.192.1-48.i.1.9, 48.192.1-48.i.1.10, 48.192.1-48.i.1.11, 48.192.1-48.i.1.12, 48.192.1-48.i.1.13, 48.192.1-48.i.1.14, 48.192.1-48.i.1.15, 48.192.1-48.i.1.16, 48.192.1-48.i.1.17, 48.192.1-48.i.1.18, 48.192.1-48.i.1.19, 48.192.1-48.i.1.20, 240.192.1-48.i.1.1, 240.192.1-48.i.1.2, 240.192.1-48.i.1.3, 240.192.1-48.i.1.4, 240.192.1-48.i.1.5, 240.192.1-48.i.1.6, 240.192.1-48.i.1.7, 240.192.1-48.i.1.8, 240.192.1-48.i.1.9, 240.192.1-48.i.1.10, 240.192.1-48.i.1.11, 240.192.1-48.i.1.12, 240.192.1-48.i.1.13, 240.192.1-48.i.1.14, 240.192.1-48.i.1.15, 240.192.1-48.i.1.16, 240.192.1-48.i.1.17, 240.192.1-48.i.1.18, 240.192.1-48.i.1.19, 240.192.1-48.i.1.20
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $12288$

Jacobian

Conductor: $2^{5}\cdot3^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 288.2.a.d

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ 2 y^{2} - z^{2} - w^{2} $
$=$ $3 x^{2} + z w$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 9 x^{4} - 2 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:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle w$

Maps to other modular curves

$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^8\,\frac{(z^{8}-z^{4}w^{4}+w^{8})^{3}}{w^{8}z^{8}(z-w)^{2}(z+w)^{2}(z^{2}+w^{2})^{2}}$

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
8.48.0.l.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.d.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bf.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bh.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1.b.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.bz.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.cb.2 $48$ $2$ $2$ $1$ $0$ 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
48.192.5.bp.1 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.bq.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.ck.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.cl.1 $48$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
48.192.5.er.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.es.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.ev.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.ew.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.ez.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.fa.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.fd.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.fe.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.288.17.ea.2 $48$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
48.384.17.hx.2 $48$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
240.192.5.bep.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.beq.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bff.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bfg.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.blr.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bls.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bmd.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bme.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bmh.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bmi.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bmt.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bmu.1 $240$ $2$ $2$ $5$ $?$ not computed