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Properties

Label 304.24.0.f.2
Level $304$
Index $24$
Genus $0$
Cusps $6$
$\Q$-cusps $2$

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Invariants

Level: $304$ $\SL_2$-level: $16$
Index: $24$ $\PSL_2$-index:$24$
Genus: $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (of which $2$ are rational) Cusp widths $1^{2}\cdot2^{3}\cdot16$ Cusp orbits $1^{2}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
$\Q$-gonality: $1$
$\overline{\Q}$-gonality: $1$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 16D0

Level structure

$\GL_2(\Z/304\Z)$-generators: $\begin{bmatrix}3&58\\208&117\end{bmatrix}$, $\begin{bmatrix}102&89\\1&262\end{bmatrix}$, $\begin{bmatrix}129&278\\302&17\end{bmatrix}$, $\begin{bmatrix}268&5\\235&134\end{bmatrix}$, $\begin{bmatrix}271&112\\86&25\end{bmatrix}$, $\begin{bmatrix}282&19\\73&156\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 304.48.0-304.f.2.1, 304.48.0-304.f.2.2, 304.48.0-304.f.2.3, 304.48.0-304.f.2.4, 304.48.0-304.f.2.5, 304.48.0-304.f.2.6, 304.48.0-304.f.2.7, 304.48.0-304.f.2.8, 304.48.0-304.f.2.9, 304.48.0-304.f.2.10, 304.48.0-304.f.2.11, 304.48.0-304.f.2.12, 304.48.0-304.f.2.13, 304.48.0-304.f.2.14, 304.48.0-304.f.2.15, 304.48.0-304.f.2.16, 304.48.0-304.f.2.17, 304.48.0-304.f.2.18, 304.48.0-304.f.2.19, 304.48.0-304.f.2.20, 304.48.0-304.f.2.21, 304.48.0-304.f.2.22, 304.48.0-304.f.2.23, 304.48.0-304.f.2.24, 304.48.0-304.f.2.25, 304.48.0-304.f.2.26, 304.48.0-304.f.2.27, 304.48.0-304.f.2.28, 304.48.0-304.f.2.29, 304.48.0-304.f.2.30, 304.48.0-304.f.2.31, 304.48.0-304.f.2.32
Cyclic 304-isogeny field degree: $40$
Cyclic 304-torsion field degree: $5760$
Full 304-torsion field degree: $126074880$

Models

This modular curve is isomorphic to $\mathbb{P}^1$.

Rational points

This modular curve has infinitely many rational points but none with conductor small enough to be contained within the database of elliptic curves over $\Q$.

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
$X_0(8)$ $8$ $2$ $2$ $0$ $0$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus
304.48.0.d.1 $304$ $2$ $2$ $0$
304.48.0.f.1 $304$ $2$ $2$ $0$
304.48.0.k.1 $304$ $2$ $2$ $0$
304.48.0.l.2 $304$ $2$ $2$ $0$
304.48.0.u.1 $304$ $2$ $2$ $0$
304.48.0.x.2 $304$ $2$ $2$ $0$
304.48.0.z.1 $304$ $2$ $2$ $0$
304.48.0.ba.2 $304$ $2$ $2$ $0$
304.48.0.bg.1 $304$ $2$ $2$ $0$
304.48.0.bh.1 $304$ $2$ $2$ $0$
304.48.0.bo.1 $304$ $2$ $2$ $0$
304.48.0.bp.2 $304$ $2$ $2$ $0$
304.48.0.bu.1 $304$ $2$ $2$ $0$
304.48.0.bv.1 $304$ $2$ $2$ $0$
304.48.0.by.1 $304$ $2$ $2$ $0$
304.48.0.bz.2 $304$ $2$ $2$ $0$
304.48.1.bi.1 $304$ $2$ $2$ $1$
304.48.1.bj.1 $304$ $2$ $2$ $1$
304.48.1.bm.1 $304$ $2$ $2$ $1$
304.48.1.bn.1 $304$ $2$ $2$ $1$
304.48.1.bs.1 $304$ $2$ $2$ $1$
304.48.1.bt.1 $304$ $2$ $2$ $1$
304.48.1.ca.1 $304$ $2$ $2$ $1$
304.48.1.cb.1 $304$ $2$ $2$ $1$