Modular curve 312.48.1.zf.1

• Label: 312.48.1.zf.1
• Level: $312$
• Index: $48$
• Genus: $1$
• Cusps: $8$
• $\Q$-cusps: $4$

Invariants

Level:	$312$		$\SL_2$-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

$\GL_2(\Z/312\Z)$-generators:	$\begin{bmatrix}31&228\\16&203\end{bmatrix}$, $\begin{bmatrix}33&59\\68&57\end{bmatrix}$, $\begin{bmatrix}49&162\\244&11\end{bmatrix}$, $\begin{bmatrix}65&267\\32&61\end{bmatrix}$, $\begin{bmatrix}125&172\\240&283\end{bmatrix}$, $\begin{bmatrix}199&131\\288&191\end{bmatrix}$, $\begin{bmatrix}289&80\\100&285\end{bmatrix}$
Contains $-I$:	yes
Quadratic refinements:	312.96.1-312.zf.1.1, 312.96.1-312.zf.1.2, 312.96.1-312.zf.1.3, 312.96.1-312.zf.1.4, 312.96.1-312.zf.1.5, 312.96.1-312.zf.1.6, 312.96.1-312.zf.1.7, 312.96.1-312.zf.1.8, 312.96.1-312.zf.1.9, 312.96.1-312.zf.1.10, 312.96.1-312.zf.1.11, 312.96.1-312.zf.1.12, 312.96.1-312.zf.1.13, 312.96.1-312.zf.1.14, 312.96.1-312.zf.1.15, 312.96.1-312.zf.1.16, 312.96.1-312.zf.1.17, 312.96.1-312.zf.1.18, 312.96.1-312.zf.1.19, 312.96.1-312.zf.1.20, 312.96.1-312.zf.1.21, 312.96.1-312.zf.1.22, 312.96.1-312.zf.1.23, 312.96.1-312.zf.1.24, 312.96.1-312.zf.1.25, 312.96.1-312.zf.1.26, 312.96.1-312.zf.1.27, 312.96.1-312.zf.1.28, 312.96.1-312.zf.1.29, 312.96.1-312.zf.1.30, 312.96.1-312.zf.1.31, 312.96.1-312.zf.1.32, 312.96.1-312.zf.1.33, 312.96.1-312.zf.1.34, 312.96.1-312.zf.1.35, 312.96.1-312.zf.1.36, 312.96.1-312.zf.1.37, 312.96.1-312.zf.1.38, 312.96.1-312.zf.1.39, 312.96.1-312.zf.1.40
Cyclic 312-isogeny field degree:	$28$
Cyclic 312-torsion field degree:	$2688$
Full 312-torsion field degree:	$40255488$

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
104.12.0.v.1	$104$	$4$	$4$	$0$	$?$	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
104.12.0.v.1	$104$	$4$	$4$	$0$	$?$	full Jacobian
312.24.0.fn.1	$312$	$2$	$2$	$0$	$?$	full Jacobian
312.24.1.hn.1	$312$	$2$	$2$	$1$	$?$	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
312.96.1.rv.1	$312$	$2$	$2$	$1$	$?$	dimension zero
312.96.1.rv.2	$312$	$2$	$2$	$1$	$?$	dimension zero
312.96.1.rv.3	$312$	$2$	$2$	$1$	$?$	dimension zero
312.96.1.rv.4	$312$	$2$	$2$	$1$	$?$	dimension zero
312.96.1.rw.1	$312$	$2$	$2$	$1$	$?$	dimension zero
312.96.1.rw.2	$312$	$2$	$2$	$1$	$?$	dimension zero
312.96.1.rw.3	$312$	$2$	$2$	$1$	$?$	dimension zero
312.96.1.rw.4	$312$	$2$	$2$	$1$	$?$	dimension zero
312.96.3.nf.1	$312$	$2$	$2$	$3$	$?$	not computed
312.96.3.nf.2	$312$	$2$	$2$	$3$	$?$	not computed
312.96.3.nh.1	$312$	$2$	$2$	$3$	$?$	not computed
312.96.3.nh.2	$312$	$2$	$2$	$3$	$?$	not computed
312.96.3.oe.1	$312$	$2$	$2$	$3$	$?$	not computed
312.96.3.of.1	$312$	$2$	$2$	$3$	$?$	not computed
312.96.3.oq.1	$312$	$2$	$2$	$3$	$?$	not computed
312.96.3.or.1	$312$	$2$	$2$	$3$	$?$	not computed
312.96.3.pc.1	$312$	$2$	$2$	$3$	$?$	not computed
312.96.3.pd.1	$312$	$2$	$2$	$3$	$?$	not computed
312.96.3.pg.1	$312$	$2$	$2$	$3$	$?$	not computed
312.96.3.ph.1	$312$	$2$	$2$	$3$	$?$	not computed
312.96.3.rb.1	$312$	$2$	$2$	$3$	$?$	not computed
312.96.3.rb.2	$312$	$2$	$2$	$3$	$?$	not computed
312.96.3.rd.1	$312$	$2$	$2$	$3$	$?$	not computed
312.96.3.rd.2	$312$	$2$	$2$	$3$	$?$	not computed
312.144.5.oh.1	$312$	$3$	$3$	$5$	$?$	not computed