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Properties

Label 48.96.1.de.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: $32$
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: $0$
$\Q$-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: 48.96.1.1838

Level structure

$\GL_2(\Z/48\Z)$-generators: $\begin{bmatrix}5&15\\20&13\end{bmatrix}$, $\begin{bmatrix}9&38\\16&21\end{bmatrix}$, $\begin{bmatrix}11&22\\12&37\end{bmatrix}$, $\begin{bmatrix}17&17\\28&17\end{bmatrix}$, $\begin{bmatrix}37&17\\8&35\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 48.192.1-48.de.1.1, 48.192.1-48.de.1.2, 48.192.1-48.de.1.3, 48.192.1-48.de.1.4, 48.192.1-48.de.1.5, 48.192.1-48.de.1.6, 48.192.1-48.de.1.7, 48.192.1-48.de.1.8, 48.192.1-48.de.1.9, 48.192.1-48.de.1.10, 48.192.1-48.de.1.11, 48.192.1-48.de.1.12, 96.192.1-48.de.1.1, 96.192.1-48.de.1.2, 96.192.1-48.de.1.3, 96.192.1-48.de.1.4, 96.192.1-48.de.1.5, 96.192.1-48.de.1.6, 96.192.1-48.de.1.7, 96.192.1-48.de.1.8, 240.192.1-48.de.1.1, 240.192.1-48.de.1.2, 240.192.1-48.de.1.3, 240.192.1-48.de.1.4, 240.192.1-48.de.1.5, 240.192.1-48.de.1.6, 240.192.1-48.de.1.7, 240.192.1-48.de.1.8, 240.192.1-48.de.1.9, 240.192.1-48.de.1.10, 240.192.1-48.de.1.11, 240.192.1-48.de.1.12
Cyclic 48-isogeny field degree: $8$
Cyclic 48-torsion field degree: $128$
Full 48-torsion field degree: $12288$

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 $ $=$ $ 3 x^{2} - z^{2} + z w - w^{2} $
$=$ $8 y^{2} + z^{2} + 2 z w - 2 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 36 x^{4} + 12 x^{2} y^{2} - 6 x^{2} z^{2} + y^{4} - 4 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{3}{2}x$
$\displaystyle Z$ $=$ $\displaystyle \frac{3}{2}w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^2}{3}\cdot\frac{(193z^{8}-1144z^{7}w+2896z^{6}w^{2}-4240z^{5}w^{3}+3976z^{4}w^{4}-2272z^{3}w^{5}+832z^{2}w^{6}-64zw^{7}+16w^{8})^{3}}{z^{2}(z-2w)^{2}(z^{2}-zw+w^{2})^{2}(z^{2}+2zw-2w^{2})^{8}}$

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.v.2 $16$ $2$ $2$ $1$ $0$ dimension zero
24.48.0.bn.1 $24$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.y.1 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bc.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.0.bf.2 $48$ $2$ $2$ $0$ $0$ full Jacobian
48.48.1.w.1 $48$ $2$ $2$ $1$ $0$ dimension zero
48.48.1.bk.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.jc.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.jd.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.je.2 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.192.5.jf.1 $48$ $2$ $2$ $5$ $0$ $2^{2}$
48.288.17.bbr.2 $48$ $3$ $3$ $17$ $1$ $1^{8}\cdot2^{4}$
48.384.17.xs.2 $48$ $4$ $4$ $17$ $0$ $1^{8}\cdot2^{4}$
96.192.5.eh.2 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.ei.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.ep.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.5.eq.1 $96$ $2$ $2$ $5$ $?$ not computed
96.192.9.gw.2 $96$ $2$ $2$ $9$ $?$ not computed
96.192.9.he.2 $96$ $2$ $2$ $9$ $?$ not computed
96.192.9.jy.2 $96$ $2$ $2$ $9$ $?$ not computed
96.192.9.kg.2 $96$ $2$ $2$ $9$ $?$ not computed
240.192.5.ccy.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.ccz.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cda.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.cdb.1 $240$ $2$ $2$ $5$ $?$ not computed