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Properties

Label 56.192.1-56.bu.1.6
Level $56$
Index $192$
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: $192$ $\PSL_2$-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: $1$
$\Q$-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: 56.192.1.632

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}3&38\\34&3\end{bmatrix}$, $\begin{bmatrix}7&24\\8&11\end{bmatrix}$, $\begin{bmatrix}9&52\\2&7\end{bmatrix}$
Contains $-I$: no $\quad$ (see 56.96.1.bu.1 for the level structure with $-I$)
Cyclic 56-isogeny field degree: $16$
Cyclic 56-torsion field degree: $384$
Full 56-torsion field degree: $16128$

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

$ 0 $ $=$ $ x^{4} - 4 x^{3} z - 3 x^{2} y^{2} - 2 x^{2} z^{2} + 6 x y^{2} z + 12 x z^{3} + 4 y^{4} + 9 y^{2} z^{2} + 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 to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{1}{2^4}\cdot\frac{17741905920xz^{23}-204031918080xz^{22}w+927014584320xz^{21}w^{2}-1878424289280xz^{20}w^{3}+504535449600xz^{19}w^{4}+4940566364160xz^{18}w^{5}-7726877245440xz^{17}w^{6}-1208805949440xz^{16}w^{7}+12458228908032xz^{15}w^{8}-6581770813440xz^{14}w^{9}-9063520026624xz^{13}w^{10}+9351866474496xz^{12}w^{11}+3249553821696xz^{11}w^{12}-6784082122752xz^{10}w^{13}+49821742080xz^{9}w^{14}+3149326047744xz^{8}w^{15}-677000956032xz^{7}w^{16}-948775947840xz^{6}w^{17}+340926513120xz^{5}w^{18}+181847113200xz^{4}w^{19}-91101352824xz^{3}w^{20}-18175444956xz^{2}w^{21}+10245035826xzw^{22}+446053257xw^{23}-8111783936z^{24}+97341407232z^{23}w-457936207872z^{22}w^{2}+932735614976z^{21}w^{3}-67238363136z^{20}w^{4}-3241424191488z^{19}w^{5}+4736747175936z^{18}w^{6}+1870547779584z^{17}w^{7}-9945436127232z^{16}w^{8}+4206477443072z^{15}w^{9}+9650465980416z^{14}w^{10}-8806122995712z^{13}w^{11}-5155188240384z^{12}w^{12}+8708104642560z^{11}w^{13}+822002058240z^{10}w^{14}-5559912850432z^{9}w^{15}+988808258688z^{8}w^{16}+2384508151296z^{7}w^{17}-878897723296z^{6}w^{18}-701360906784z^{5}w^{19}+405987229992z^{4}w^{20}+127650612624z^{3}w^{21}-99501928518z^{2}w^{22}-10245035826zw^{23}+16733815927w^{24}}{w^{8}(3047424xz^{15}-22855680xz^{14}w+58662912xz^{13}w^{2}-34664448xz^{12}w^{3}-96565248xz^{11}w^{4}+149799936xz^{10}w^{5}+18332160xz^{9}w^{6}-143776512xz^{8}w^{7}+39283584xz^{7}w^{8}+56828352xz^{6}w^{9}-22426272xz^{5}w^{10}-9224400xz^{4}w^{11}+3077592xz^{3}w^{12}+452844xz^{2}w^{13}+26622xzw^{14}+567xw^{15}-2916352z^{16}+23330816z^{15}w-63397888z^{14}w^{2}+35495936z^{13}w^{3}+135596032z^{12}w^{4}-213471232z^{11}w^{5}-49544704z^{10}w^{6}+270274048z^{9}w^{7}-67103360z^{8}w^{8}-147216896z^{7}w^{9}+59272160z^{6}w^{10}+35792224z^{5}w^{11}-12561448z^{4}w^{12}-3070032z^{3}w^{13}-452682z^{2}w^{14}-26622zw^{15}-567w^{16})}$

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

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}w$

Equation of the image curve:

$0$ $=$ $ X^{4}-3X^{2}Y^{2}+4Y^{4}-4X^{3}Z+6XY^{2}Z-2X^{2}Z^{2}+9Y^{2}Z^{2}+12XZ^{3}+2Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.96.0-8.h.1.4 $8$ $2$ $2$ $0$ $0$ full Jacobian
56.96.0-56.g.1.6 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.0-56.g.1.16 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.0-8.h.1.2 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.0-56.i.2.4 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.0-56.i.2.7 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.0-56.w.1.2 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.0-56.w.1.16 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.96.1-56.bg.2.12 $56$ $2$ $2$ $1$ $1$ dimension zero
56.96.1-56.bg.2.14 $56$ $2$ $2$ $1$ $1$ dimension zero
56.96.1-56.bi.2.8 $56$ $2$ $2$ $1$ $1$ dimension zero
56.96.1-56.bi.2.10 $56$ $2$ $2$ $1$ $1$ dimension zero
56.96.1-56.bs.1.9 $56$ $2$ $2$ $1$ $1$ dimension zero
56.96.1-56.bs.1.12 $56$ $2$ $2$ $1$ $1$ 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
56.1536.49-56.no.2.14 $56$ $8$ $8$ $49$ $8$ $1^{20}\cdot2^{6}\cdot4^{4}$
56.4032.145-56.bkg.2.6 $56$ $21$ $21$ $145$ $24$ $1^{16}\cdot2^{26}\cdot4\cdot6^{4}\cdot12^{4}$
56.5376.193-56.bli.1.9 $56$ $28$ $28$ $193$ $31$ $1^{36}\cdot2^{32}\cdot4^{5}\cdot6^{4}\cdot12^{4}$