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Properties

Label 56.96.1.bu.2
Level $56$
Index $96$
Genus $1$
Analytic rank $1$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $56$ $\SL_2$-level: $8$ Newform level: $3136$
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 $4^{8}\cdot8^{8}$ Cusp orbits $2^{2}\cdot4^{3}$
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: 8K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.96.1.352

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}5&26\\16&55\end{bmatrix}$, $\begin{bmatrix}17&34\\6&49\end{bmatrix}$, $\begin{bmatrix}25&0\\30&43\end{bmatrix}$, $\begin{bmatrix}47&40\\26&53\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 56.192.1-56.bu.2.1, 56.192.1-56.bu.2.2, 56.192.1-56.bu.2.3, 56.192.1-56.bu.2.4, 56.192.1-56.bu.2.5, 56.192.1-56.bu.2.6, 56.192.1-56.bu.2.7, 56.192.1-56.bu.2.8, 168.192.1-56.bu.2.1, 168.192.1-56.bu.2.2, 168.192.1-56.bu.2.3, 168.192.1-56.bu.2.4, 168.192.1-56.bu.2.5, 168.192.1-56.bu.2.6, 168.192.1-56.bu.2.7, 168.192.1-56.bu.2.8, 280.192.1-56.bu.2.1, 280.192.1-56.bu.2.2, 280.192.1-56.bu.2.3, 280.192.1-56.bu.2.4, 280.192.1-56.bu.2.5, 280.192.1-56.bu.2.6, 280.192.1-56.bu.2.7, 280.192.1-56.bu.2.8
Cyclic 56-isogeny field degree: $16$
Cyclic 56-torsion field degree: $192$
Full 56-torsion field degree: $32256$

Jacobian

Conductor: $2^{6}\cdot7^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 3136.2.a.m

Models

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

$ 0 $ $=$ $ 2 x^{2} - 2 y^{2} - z^{2} $
$=$ $4 x^{2} + 3 y^{2} + 5 z^{2} - w^{2}$
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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 to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{7^4}\cdot\frac{(2401z^{8}-1372z^{6}w^{2}+980z^{4}w^{4}-224z^{2}w^{6}+16w^{8})^{3}}{w^{4}z^{8}(7z^{2}-2w^{2})^{4}(7z^{2}-w^{2})^{2}}$

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.h.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
56.48.0.g.2 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.48.0.i.1 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.48.0.w.1 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.48.1.bg.1 $56$ $2$ $2$ $1$ $1$ dimension zero
56.48.1.bi.1 $56$ $2$ $2$ $1$ $1$ dimension zero
56.48.1.bs.1 $56$ $2$ $2$ $1$ $1$ 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
56.768.49.no.1 $56$ $8$ $8$ $49$ $8$ $1^{20}\cdot2^{6}\cdot4^{4}$
56.2016.145.bkg.1 $56$ $21$ $21$ $145$ $24$ $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$
56.2688.193.bli.2 $56$ $28$ $28$ $193$ $31$ $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$
168.288.17.jbw.1 $168$ $3$ $3$ $17$ $?$ not computed
168.384.17.drm.1 $168$ $4$ $4$ $17$ $?$ not computed