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Properties

Label 48.96.1.br.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: $64$
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: 48.96.1.2164

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}13&7\\16&9\end{bmatrix}$, $\begin{bmatrix}15&47\\8&37\end{bmatrix}$, $\begin{bmatrix}19&18\\16&35\end{bmatrix}$, $\begin{bmatrix}33&43\\32&5\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.br.1.1, 48.192.1-48.br.1.2, 48.192.1-48.br.1.3, 48.192.1-48.br.1.4, 48.192.1-48.br.1.5, 48.192.1-48.br.1.6, 48.192.1-48.br.1.7, 48.192.1-48.br.1.8, 96.192.1-48.br.1.1, 96.192.1-48.br.1.2, 96.192.1-48.br.1.3, 96.192.1-48.br.1.4, 240.192.1-48.br.1.1, 240.192.1-48.br.1.2, 240.192.1-48.br.1.3, 240.192.1-48.br.1.4, 240.192.1-48.br.1.5, 240.192.1-48.br.1.6, 240.192.1-48.br.1.7, 240.192.1-48.br.1.8
Cyclic 48-isogeny field degree: $4$
Cyclic 48-torsion field degree: $64$
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 $ $=$ $ x^{2} - 2 x z - 2 z^{2} + 2 w^{2} $
$=$ $x^{2} + x z - 3 y^{2} + z^{2} + 3 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 4 x^{2} y^{2} + 30 x^{2} z^{2} + y^{4} - 6 y^{2} z^{2} + 9 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:

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{3}w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^8}{3}\cdot\frac{1120043793960xz^{23}-2906367924204xz^{21}w^{2}+3041356063968xz^{19}w^{4}-1630915192908xz^{17}w^{6}+466011558720xz^{15}w^{8}-64372018032xz^{13}w^{10}+2347193376xz^{11}w^{12}+240441696xz^{9}w^{14}-9758880xz^{7}w^{16}-483408xz^{5}w^{18}+1728xz^{3}w^{20}+144xzw^{22}+819928963881z^{24}-2774266571952z^{22}w^{2}+3796643201814z^{20}w^{4}-2706100563816z^{18}w^{6}+1068351708870z^{16}w^{8}-224935224480z^{14}w^{10}+20340348048z^{12}w^{12}+129587040z^{10}w^{14}-91205676z^{8}w^{16}-354240z^{6}w^{18}+109080z^{4}w^{20}+1440z^{2}w^{22}-8w^{24}}{w^{16}(1512xz^{7}-972xz^{5}w^{2}+96xz^{3}w^{4}+4xzw^{6}+1107z^{8}-1584z^{6}w^{2}+486z^{4}w^{4}-8z^{2}w^{6}-w^{8})}$

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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.i.1 $16$ $2$ $2$ $1$ $0$ dimension zero
24.48.0.bf.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.l.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bs.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bu.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1.bg.2 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.bi.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
48.288.17.om.2 $48$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
48.384.17.px.2 $48$ $4$ $4$ $17$ $1$ $1^{8}\cdot2^{4}$
96.192.5.by.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.cb.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.cf.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.ci.1 $96$ $2$ $2$ $5$ $?$ not computed