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Properties

Label 56.192.1-56.bc.1.7
Level $56$
Index $192$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $56$ $\SL_2$-level: $8$ Newform level: $64$
Index: $192$ $\PSL_2$-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: $0$
$\Q$-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: 56.192.1.804

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}1&20\\40&13\end{bmatrix}$, $\begin{bmatrix}31&48\\32&33\end{bmatrix}$, $\begin{bmatrix}41&24\\18&53\end{bmatrix}$
Contains $-I$: no $\quad$ (see 56.96.1.bc.1 for the level structure with $-I$)
Cyclic 56-isogeny field degree: $16$
Cyclic 56-torsion field degree: $192$
Full 56-torsion field degree: $16128$

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 $ $=$ $ 3 y^{2} - 2 y z - 2 z^{2} - w^{2} $
$=$ $7 x^{2} - 2 y^{2} - y z - z^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 8 x^{2} y^{2} + 21 x^{2} z^{2} + 9 y^{4} - 84 y^{2} z^{2} + 196 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 to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{3^8\cdot7^2}\cdot\frac{79521843187572655063040yz^{23}+187444344656421258362880yz^{21}w^{2}+191704443398612650598400yz^{19}w^{4}+111501564017560419225600yz^{17}w^{6}+40636703517444258545664yz^{15}w^{8}+9647568675604751302656yz^{13}w^{10}+1505198356054231168512yz^{11}w^{12}+152398958704620122880yz^{9}w^{14}+9682919279551784448yz^{7}w^{16}+362156500016349696yz^{5}w^{18}+7017805057074336yz^{3}w^{20}+50993583603984yzw^{22}+43624393280518367916032z^{24}+117857142949016707145728z^{22}w^{2}+138979433930930883892224z^{20}w^{4}+93946328531977601111040z^{18}w^{6}+40203679377737792383488z^{16}w^{8}+11356081099301645586432z^{14}w^{10}+2143565966466311429952z^{12}w^{12}+268278985638485857728z^{10}w^{14}+21687535695906648288z^{8}w^{16}+1078261881434785152z^{6}w^{18}+30131068675730148z^{4}w^{20}+390385041919236z^{2}w^{22}+1273519880379w^{24}}{w^{8}(1353569627648yz^{15}+2030354441472yz^{13}w^{2}+1211997294144yz^{11}w^{4}+366262918560yz^{9}w^{6}+58834106352yz^{7}w^{8}+4797105768yz^{5}w^{10}+168608952yz^{3}w^{12}+1609632yzw^{14}+742553428400z^{16}+1369632279232z^{14}w^{2}+1021184234616z^{12}w^{4}+394737085512z^{10}w^{6}+84188852307z^{8}w^{8}+9705045294z^{6}w^{10}+544565187z^{4}w^{12}+11331576z^{2}w^{14}+34992w^{16})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 56.96.1.bc.1 :

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle x$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{7}w$

Equation of the image curve:

$0$ $=$ $ X^{4}-8X^{2}Y^{2}+9Y^{4}+21X^{2}Z^{2}-84Y^{2}Z^{2}+196Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.96.1-8.m.2.2 $8$ $2$ $2$ $1$ $0$ dimension zero
56.96.0-56.g.1.1 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.0-56.g.1.12 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.0-56.i.2.8 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.0-56.i.2.13 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.0-56.o.1.7 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.0-56.o.1.9 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.0-56.q.1.2 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.0-56.q.1.11 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.1-8.m.2.7 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1-56.u.1.13 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1-56.u.1.14 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1-56.w.1.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1-56.w.1.15 $56$ $2$ $2$ $1$ $0$ 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
56.1536.49-56.gg.2.8 $56$ $8$ $8$ $49$ $8$ $1^{20}\cdot2^{6}\cdot4^{4}$
56.4032.145-56.pu.2.1 $56$ $21$ $21$ $145$ $23$ $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$
56.5376.193-56.qo.2.5 $56$ $28$ $28$ $193$ $31$ $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$