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Properties

Label 68.12.0.a.1
Level $68$
Index $12$
Genus $0$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $68$ $\SL_2$-level: $4$
Index: $12$ $\PSL_2$-index:$12$
Genus: $0 = 1 + \frac{ 12 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (of which $2$ are rational) Cusp widths $2^{2}\cdot4^{2}$ Cusp orbits $1^{2}\cdot2$
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: 4E0
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 68.12.0.3

Level structure

$\GL_2(\Z/68\Z)$-generators: $\begin{bmatrix}5&24\\54&47\end{bmatrix}$, $\begin{bmatrix}17&26\\32&7\end{bmatrix}$, $\begin{bmatrix}29&12\\24&59\end{bmatrix}$, $\begin{bmatrix}43&48\\32&41\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 68.24.0-68.a.1.1, 68.24.0-68.a.1.2, 68.24.0-68.a.1.3, 68.24.0-68.a.1.4, 136.24.0-68.a.1.1, 136.24.0-68.a.1.2, 136.24.0-68.a.1.3, 136.24.0-68.a.1.4, 204.24.0-68.a.1.1, 204.24.0-68.a.1.2, 204.24.0-68.a.1.3, 204.24.0-68.a.1.4
Cyclic 68-isogeny field degree: $36$
Cyclic 68-torsion field degree: $1152$
Full 68-torsion field degree: $626688$

Models

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

Rational points

This modular curve has infinitely many rational points, including 178 stored non-cuspidal points.

Maps to other modular curves

$j$-invariant map of degree 12 to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{1}{2^{16}\cdot17^2}\cdot\frac{x^{12}(289x^{4}+69632x^{2}y^{2}+16777216y^{4})^{3}}{y^{4}x^{16}(17x^{2}+4096y^{2})^{2}}$

Modular covers

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Cover information

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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
$X(2)$ $2$ $2$ $2$ $0$ $0$
68.6.0.b.1 $68$ $2$ $2$ $0$ $0$
68.6.0.e.1 $68$ $2$ $2$ $0$ $0$

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

Covering curve Level Index Degree Genus
68.24.0.a.1 $68$ $2$ $2$ $0$
68.24.0.c.1 $68$ $2$ $2$ $0$
68.216.15.c.1 $68$ $18$ $18$ $15$
68.1632.121.c.1 $68$ $136$ $136$ $121$
68.1836.136.c.1 $68$ $153$ $153$ $136$
136.24.0.b.1 $136$ $2$ $2$ $0$
136.24.0.f.1 $136$ $2$ $2$ $0$
204.24.0.d.1 $204$ $2$ $2$ $0$
204.24.0.f.1 $204$ $2$ $2$ $0$
204.36.2.a.1 $204$ $3$ $3$ $2$
204.48.1.a.1 $204$ $4$ $4$ $1$