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Properties

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

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Invariants

Level: $48$ $\SL_2$-level: $16$ Newform level: $576$
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 $2^{4}\cdot4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $1$
$\Q$-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: 48.96.1.1286

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}5&40\\4&7\end{bmatrix}$, $\begin{bmatrix}9&29\\44&17\end{bmatrix}$, $\begin{bmatrix}19&38\\12&17\end{bmatrix}$, $\begin{bmatrix}25&0\\44&35\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.u.1.1, 48.192.1-48.u.1.2, 48.192.1-48.u.1.3, 48.192.1-48.u.1.4, 48.192.1-48.u.1.5, 48.192.1-48.u.1.6, 48.192.1-48.u.1.7, 48.192.1-48.u.1.8, 240.192.1-48.u.1.1, 240.192.1-48.u.1.2, 240.192.1-48.u.1.3, 240.192.1-48.u.1.4, 240.192.1-48.u.1.5, 240.192.1-48.u.1.6, 240.192.1-48.u.1.7, 240.192.1-48.u.1.8
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $64$
Full 48-torsion field degree: $12288$

Jacobian

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

Models

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

$ 0 $ $=$ $ x^{2} + y w - z^{2} - w^{2} $
$=$ $6 x^{2} - y^{2} - 2 y w + 2 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 2 x^{2} y^{2} + 4 x^{2} z^{2} + y^{4} - 10 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:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 2^8\,\frac{144yz^{22}w+41520yz^{20}w^{3}+3116592yz^{18}w^{5}+96391536yz^{16}w^{7}+1464837792yz^{14}w^{9}+11552088096yz^{12}w^{11}+44345381040yz^{10}w^{13}+50681545680yz^{8}w^{15}-116061939180yz^{6}w^{17}-167314531620yz^{4}w^{19}+272815492164yz^{2}w^{21}-90354434940yw^{23}-8z^{24}-6912z^{22}w^{2}-851256z^{20}w^{4}-36268128z^{18}w^{6}-706137612z^{16}w^{8}-6994009632z^{14}w^{10}-35208948400z^{12}w^{12}-74327424480z^{10}w^{14}+17735052006z^{8}w^{16}+217669922640z^{6}w^{18}-20110516206z^{4}w^{20}-189924965400z^{2}w^{22}+81226783441w^{24}}{z^{2}(40yz^{20}w+7536yz^{18}w^{3}+445848yz^{16}w^{5}+12710632yz^{14}w^{7}+207188552yz^{12}w^{9}+2096059224yz^{10}w^{11}+13646702488yz^{8}w^{13}+57358628408yz^{6}w^{15}+150619663728yz^{4}w^{17}+224968712280yz^{2}w^{19}+146042423984yw^{21}-3z^{22}-1480z^{20}w^{2}-131304z^{18}w^{4}-4831328z^{16}w^{6}-95380258z^{14}w^{8}-1135191552z^{12}w^{10}-8617399604z^{10}w^{12}-42595356304z^{8}w^{14}-135887960979z^{6}w^{16}-267579103928z^{4}w^{18}-291674612124z^{2}w^{20}-131289143184w^{22})}$

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
16.48.0.e.1 $16$ $2$ $2$ $0$ $0$ full Jacobian
24.48.0.bd.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bc.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bd.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1.d.1 $48$ $2$ $2$ $1$ $1$ dimension zero
48.48.1.bo.1 $48$ $2$ $2$ $1$ $1$ dimension zero
48.48.1.bp.2 $48$ $2$ $2$ $1$ $1$ 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
48.288.17.gm.2 $48$ $3$ $3$ $17$ $2$ $1^{8}\cdot2^{4}$
48.384.17.ju.2 $48$ $4$ $4$ $17$ $2$ $1^{8}\cdot2^{4}$