↑

Properties

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

Related objects

Downloads

Learn more

Invariants

Level: $56$ $\SL_2$-level: $8$ Newform level: $64$
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: $0$
$\Q$-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: 56.48.1.280

Level structure

$\GL_2(\Z/56\Z)$-generators: $\begin{bmatrix}15&12\\36&31\end{bmatrix}$, $\begin{bmatrix}23&44\\30&29\end{bmatrix}$, $\begin{bmatrix}25&36\\18&23\end{bmatrix}$, $\begin{bmatrix}25&46\\32&15\end{bmatrix}$, $\begin{bmatrix}25&48\\54&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 56.96.1-56.x.2.1, 56.96.1-56.x.2.2, 56.96.1-56.x.2.3, 56.96.1-56.x.2.4, 56.96.1-56.x.2.5, 56.96.1-56.x.2.6, 56.96.1-56.x.2.7, 56.96.1-56.x.2.8, 56.96.1-56.x.2.9, 56.96.1-56.x.2.10, 56.96.1-56.x.2.11, 56.96.1-56.x.2.12, 56.96.1-56.x.2.13, 56.96.1-56.x.2.14, 56.96.1-56.x.2.15, 56.96.1-56.x.2.16, 168.96.1-56.x.2.1, 168.96.1-56.x.2.2, 168.96.1-56.x.2.3, 168.96.1-56.x.2.4, 168.96.1-56.x.2.5, 168.96.1-56.x.2.6, 168.96.1-56.x.2.7, 168.96.1-56.x.2.8, 168.96.1-56.x.2.9, 168.96.1-56.x.2.10, 168.96.1-56.x.2.11, 168.96.1-56.x.2.12, 168.96.1-56.x.2.13, 168.96.1-56.x.2.14, 168.96.1-56.x.2.15, 168.96.1-56.x.2.16, 280.96.1-56.x.2.1, 280.96.1-56.x.2.2, 280.96.1-56.x.2.3, 280.96.1-56.x.2.4, 280.96.1-56.x.2.5, 280.96.1-56.x.2.6, 280.96.1-56.x.2.7, 280.96.1-56.x.2.8, 280.96.1-56.x.2.9, 280.96.1-56.x.2.10, 280.96.1-56.x.2.11, 280.96.1-56.x.2.12, 280.96.1-56.x.2.13, 280.96.1-56.x.2.14, 280.96.1-56.x.2.15, 280.96.1-56.x.2.16
Cyclic 56-isogeny field degree: $16$
Cyclic 56-torsion field degree: $384$
Full 56-torsion field degree: $64512$

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 $ $=$ $ 7 x^{2} - 7 x y + w^{2} $
$=$ $7 x^{2} + 7 x y + 14 y^{2} - z^{2} + w^{2}$
 Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} + 14 x^{2} y^{2} - 21 x^{2} z^{2} + 98 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{14}w$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{14}z$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -2^2\,\frac{882y^{2}z^{10}-5292y^{2}z^{8}w^{2}+1008y^{2}z^{6}w^{4}+2016y^{2}z^{4}w^{6}-42336y^{2}z^{2}w^{8}+28224y^{2}w^{10}-31z^{12}+120z^{10}w^{2}+192z^{8}w^{4}-512z^{6}w^{6}+4080z^{4}w^{8}-6144z^{2}w^{10}+2048w^{12}}{w^{4}z^{4}(14y^{2}z^{2}+28y^{2}w^{2}-z^{4})}$

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
8.24.1.d.1 $8$ $2$ $2$ $1$ $0$ dimension zero
56.24.0.h.1 $56$ $2$ $2$ $0$ $0$ full Jacobian
56.24.0.i.1 $56$ $2$ $2$ $0$ $0$ full Jacobian

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.e.2 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.p.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.ba.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.bg.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.bi.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.bo.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.bq.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.96.1.br.1 $56$ $2$ $2$ $1$ $0$ dimension zero
56.384.25.cj.2 $56$ $8$ $8$ $25$ $3$ $1^{8}\cdot2^{4}\cdot4^{2}$
56.1008.73.ed.2 $56$ $21$ $21$ $73$ $9$ $1^{4}\cdot2^{14}\cdot4\cdot6^{2}\cdot12^{2}$
56.1344.97.ed.1 $56$ $28$ $28$ $97$ $12$ $1^{12}\cdot2^{18}\cdot4^{3}\cdot6^{2}\cdot12^{2}$
168.96.1.de.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.dg.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.du.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ee.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.ek.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.eu.2 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.fi.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.96.1.fk.1 $168$ $2$ $2$ $1$ $?$ dimension zero
168.144.9.ep.2 $168$ $3$ $3$ $9$ $?$ not computed
168.192.9.cv.2 $168$ $4$ $4$ $9$ $?$ not computed
280.96.1.de.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.dg.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.du.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.ee.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.ek.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.eu.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.fi.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.fk.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.240.17.bv.1 $280$ $5$ $5$ $17$ $?$ not computed
280.288.17.df.2 $280$ $6$ $6$ $17$ $?$ not computed