↑

Properties

Label 56.48.1.bc.2
Level $56$
Index $48$
Genus $1$
Analytic rank $1$
Cusps $8$
$\Q$-cusps $0$

Related objects

Downloads

Learn more

Invariants

Level: $56$ $\SL_2$-level: $8$ Newform level: $1568$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $2^{4}$
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: 8G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 56.48.1.52

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}13&48\\46&5\end{bmatrix}$, $\begin{bmatrix}23&48\\32&1\end{bmatrix}$, $\begin{bmatrix}37&4\\54&15\end{bmatrix}$, $\begin{bmatrix}49&4\\22&55\end{bmatrix}$, $\begin{bmatrix}49&12\\18&9\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 56.96.1-56.bc.2.1, 56.96.1-56.bc.2.2, 56.96.1-56.bc.2.3, 56.96.1-56.bc.2.4, 56.96.1-56.bc.2.5, 56.96.1-56.bc.2.6, 56.96.1-56.bc.2.7, 56.96.1-56.bc.2.8, 56.96.1-56.bc.2.9, 56.96.1-56.bc.2.10, 56.96.1-56.bc.2.11, 56.96.1-56.bc.2.12, 56.96.1-56.bc.2.13, 56.96.1-56.bc.2.14, 56.96.1-56.bc.2.15, 56.96.1-56.bc.2.16, 168.96.1-56.bc.2.1, 168.96.1-56.bc.2.2, 168.96.1-56.bc.2.3, 168.96.1-56.bc.2.4, 168.96.1-56.bc.2.5, 168.96.1-56.bc.2.6, 168.96.1-56.bc.2.7, 168.96.1-56.bc.2.8, 168.96.1-56.bc.2.9, 168.96.1-56.bc.2.10, 168.96.1-56.bc.2.11, 168.96.1-56.bc.2.12, 168.96.1-56.bc.2.13, 168.96.1-56.bc.2.14, 168.96.1-56.bc.2.15, 168.96.1-56.bc.2.16, 280.96.1-56.bc.2.1, 280.96.1-56.bc.2.2, 280.96.1-56.bc.2.3, 280.96.1-56.bc.2.4, 280.96.1-56.bc.2.5, 280.96.1-56.bc.2.6, 280.96.1-56.bc.2.7, 280.96.1-56.bc.2.8, 280.96.1-56.bc.2.9, 280.96.1-56.bc.2.10, 280.96.1-56.bc.2.11, 280.96.1-56.bc.2.12, 280.96.1-56.bc.2.13, 280.96.1-56.bc.2.14, 280.96.1-56.bc.2.15, 280.96.1-56.bc.2.16
Cyclic 56-isogeny field degree: $16$
Cyclic 56-torsion field degree: $384$
Full 56-torsion field degree: $64512$

Jacobian

Conductor: $2^{5}\cdot7^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 1568.2.a.e

Models

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

$ 0 $ $=$ $ x z + y^{2} + z^{2} $
$=$ $7 x^{2} - 5 x z + 2 y^{2} + 2 z^{2} - w^{2}$
 Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} + 3 x^{2} z^{2} - 7 y^{2} z^{2} + 2 z^{4} $
 Copy content Toggle raw display

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:

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

Maps to other modular curves

$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{2^8}{7^2}\cdot\frac{100842xz^{9}w^{2}+28812xz^{7}w^{4}+6860xz^{5}w^{6}+980xz^{3}w^{8}+42xzw^{10}+117649z^{12}+7203z^{8}w^{4}+4116z^{6}w^{6}+539z^{4}w^{8}+84z^{2}w^{10}+w^{12}}{w^{4}z^{4}(196xz^{3}+28xzw^{2}+28z^{2}w^{2}+w^{4})}$

Modular covers

Sorry, your browser does not support the nearby lattice.

Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.24.0.d.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
56.24.0.h.2 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.24.1.c.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
56.96.1.b.2 $56$ $2$ $2$ $1$ $1$ dimension zero
56.96.1.h.1 $56$ $2$ $2$ $1$ $1$ dimension zero
56.96.1.bb.2 $56$ $2$ $2$ $1$ $1$ dimension zero
56.96.1.bd.1 $56$ $2$ $2$ $1$ $1$ dimension zero
56.96.1.bt.2 $56$ $2$ $2$ $1$ $1$ dimension zero
56.96.1.bv.1 $56$ $2$ $2$ $1$ $1$ dimension zero
56.96.1.ca.1 $56$ $2$ $2$ $1$ $1$ dimension zero
56.96.1.cb.2 $56$ $2$ $2$ $1$ $1$ dimension zero
56.384.25.dq.2 $56$ $8$ $8$ $25$ $4$ $1^{8}\cdot2^{4}\cdot4^{2}$
56.1008.73.gq.1 $56$ $21$ $21$ $73$ $10$ $1^{4}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.1344.97.gq.1 $56$ $28$ $28$ $97$ $13$ $1^{12}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
168.96.1.fq.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.fu.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.gx.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.hb.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.lz.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.md.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.nf.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.nj.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.144.9.rg.2 $168$ $3$ $3$ $9$ $?$ not computed
168.192.9.je.2 $168$ $4$ $4$ $9$ $?$ not computed
280.96.1.fq.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.fu.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.gx.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.hb.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.lf.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.lj.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.ml.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.mp.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.240.17.ei.2 $280$ $5$ $5$ $17$ $?$ not computed
280.288.17.le.1 $280$ $6$ $6$ $17$ $?$ not computed