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Properties

Label 48.96.1.bl.2
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: $64$
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 $2^{2}\cdot4^{3}$
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: 48.96.1.1629

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}3&34\\8&11\end{bmatrix}$, $\begin{bmatrix}7&24\\16&7\end{bmatrix}$, $\begin{bmatrix}11&2\\40&39\end{bmatrix}$, $\begin{bmatrix}11&39\\16&41\end{bmatrix}$, $\begin{bmatrix}35&44\\32&15\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.bl.2.1, 48.192.1-48.bl.2.2, 48.192.1-48.bl.2.3, 48.192.1-48.bl.2.4, 48.192.1-48.bl.2.5, 48.192.1-48.bl.2.6, 48.192.1-48.bl.2.7, 48.192.1-48.bl.2.8, 48.192.1-48.bl.2.9, 48.192.1-48.bl.2.10, 48.192.1-48.bl.2.11, 48.192.1-48.bl.2.12, 240.192.1-48.bl.2.1, 240.192.1-48.bl.2.2, 240.192.1-48.bl.2.3, 240.192.1-48.bl.2.4, 240.192.1-48.bl.2.5, 240.192.1-48.bl.2.6, 240.192.1-48.bl.2.7, 240.192.1-48.bl.2.8, 240.192.1-48.bl.2.9, 240.192.1-48.bl.2.10, 240.192.1-48.bl.2.11, 240.192.1-48.bl.2.12
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $12288$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

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

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

$ 0 $ $=$ $ x^{4} - x^{2} y^{2} + 12 x^{2} z^{2} + y^{4} - 60 y^{2} z^{2} + 900 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle x$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{6}w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{1}{3}\cdot\frac{8707129344yz^{21}w^{2}-15237476352yz^{19}w^{4}+15287864832yz^{15}w^{8}-10887270912yz^{13}w^{10}+1649382912yz^{11}w^{12}+929947392yz^{9}w^{14}-369915984yz^{7}w^{16}+29793744yz^{5}w^{18}+2104488yz^{3}w^{20}+504yzw^{22}-2176782336z^{24}+15237476352z^{20}w^{4}-22211241984z^{18}w^{6}+7538116608z^{16}w^{8}+6353987328z^{14}w^{10}-6327766656z^{12}w^{12}+1894793472z^{10}w^{14}-67639536z^{8}w^{16}-69194952z^{6}w^{18}+10011708z^{4}w^{20}-41796z^{2}w^{22}-w^{24}}{w^{16}(432yz^{7}-432yz^{5}w^{2}-24yz^{3}w^{4}+56yzw^{6}+504z^{6}w^{2}-372z^{4}w^{4}+60z^{2}w^{6}+5w^{8})}$

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
16.48.1.g.1 $16$ $2$ $2$ $1$ $0$ dimension zero
24.48.0.be.2 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.k.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.by.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bz.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1.bi.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.bj.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.hj.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.hk.1 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.hl.2 $48$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
48.192.5.hm.2 $48$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
48.288.17.ns.2 $48$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
48.384.17.pk.2 $48$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
240.192.5.bye.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.byf.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.byg.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.byh.2 $240$ $2$ $2$ $5$ $?$ not computed