↑

Properties

Label 156.48.1.k.1
Level $156$
Index $48$
Genus $1$
Cusps $8$
$\Q$-cusps $4$

Related objects

Downloads

Learn more

Invariants

Level: $156$ $\SL_2$-level: $12$ Newform level: $1$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (of which $4$ are rational) Cusp widths $2^{2}\cdot4^{2}\cdot6^{2}\cdot12^{2}$ Cusp orbits $1^{4}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 12P1

Level structure

$\GL_2(\Z/156\Z)$-generators: $\begin{bmatrix}67&107\\136&129\end{bmatrix}$, $\begin{bmatrix}93&28\\68&49\end{bmatrix}$, $\begin{bmatrix}111&92\\80&123\end{bmatrix}$, $\begin{bmatrix}133&24\\28&143\end{bmatrix}$, $\begin{bmatrix}133&132\\4&89\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 156.96.1-156.k.1.1, 156.96.1-156.k.1.2, 156.96.1-156.k.1.3, 156.96.1-156.k.1.4, 156.96.1-156.k.1.5, 156.96.1-156.k.1.6, 156.96.1-156.k.1.7, 156.96.1-156.k.1.8, 156.96.1-156.k.1.9, 156.96.1-156.k.1.10, 156.96.1-156.k.1.11, 156.96.1-156.k.1.12, 312.96.1-156.k.1.1, 312.96.1-156.k.1.2, 312.96.1-156.k.1.3, 312.96.1-156.k.1.4, 312.96.1-156.k.1.5, 312.96.1-156.k.1.6, 312.96.1-156.k.1.7, 312.96.1-156.k.1.8, 312.96.1-156.k.1.9, 312.96.1-156.k.1.10, 312.96.1-156.k.1.11, 312.96.1-156.k.1.12, 312.96.1-156.k.1.13, 312.96.1-156.k.1.14, 312.96.1-156.k.1.15, 312.96.1-156.k.1.16, 312.96.1-156.k.1.17, 312.96.1-156.k.1.18, 312.96.1-156.k.1.19, 312.96.1-156.k.1.20, 312.96.1-156.k.1.21, 312.96.1-156.k.1.22, 312.96.1-156.k.1.23, 312.96.1-156.k.1.24, 312.96.1-156.k.1.25, 312.96.1-156.k.1.26, 312.96.1-156.k.1.27, 312.96.1-156.k.1.28
Cyclic 156-isogeny field degree: $14$
Cyclic 156-torsion field degree: $672$
Full 156-torsion field degree: $2515968$

Jacobian

Conductor: $?$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: not computed

Rational points

This modular curve is an elliptic curve, but the rank has not been computed

Modular covers

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_0(3)$ $3$ $12$ $12$ $0$ $0$ full Jacobian
52.12.0.g.1 $52$ $4$ $4$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
$X_0(12)$ $12$ $2$ $2$ $0$ $0$ full Jacobian
52.12.0.g.1 $52$ $4$ $4$ $0$ $0$ full Jacobian
78.24.0.b.1 $78$ $2$ $2$ $0$ $?$ full Jacobian
156.24.1.o.1 $156$ $2$ $2$ $1$ $?$ 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
156.96.1.l.1 $156$ $2$ $2$ $1$ $?$ dimension zero
156.96.1.l.2 $156$ $2$ $2$ $1$ $?$ dimension zero
156.96.1.l.3 $156$ $2$ $2$ $1$ $?$ dimension zero
156.96.1.l.4 $156$ $2$ $2$ $1$ $?$ dimension zero
156.144.5.ca.1 $156$ $3$ $3$ $5$ $?$ not computed
312.96.1.rr.1 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.rr.2 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.rr.3 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.1.rr.4 $312$ $2$ $2$ $1$ $?$ dimension zero
312.96.3.na.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.na.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.nc.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.nc.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ny.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.nz.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ok.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ol.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.os.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ot.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ow.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.ox.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.qw.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.qw.2 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.qy.1 $312$ $2$ $2$ $3$ $?$ not computed
312.96.3.qy.2 $312$ $2$ $2$ $3$ $?$ not computed