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Properties

Label 28.12.0.a.1
Level $28$
Index $12$
Genus $0$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $28$ $\SL_2$-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: 28.12.0.3

Level structure

$\GL_2(\Z/28\Z)$-generators: $\begin{bmatrix}1&2\\8&21\end{bmatrix}$, $\begin{bmatrix}7&22\\6&19\end{bmatrix}$, $\begin{bmatrix}11&4\\12&21\end{bmatrix}$, $\begin{bmatrix}17&22\\4&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 28.24.0-28.a.1.1, 28.24.0-28.a.1.2, 28.24.0-28.a.1.3, 28.24.0-28.a.1.4, 56.24.0-28.a.1.1, 56.24.0-28.a.1.2, 56.24.0-28.a.1.3, 56.24.0-28.a.1.4, 84.24.0-28.a.1.1, 84.24.0-28.a.1.2, 84.24.0-28.a.1.3, 84.24.0-28.a.1.4, 140.24.0-28.a.1.1, 140.24.0-28.a.1.2, 140.24.0-28.a.1.3, 140.24.0-28.a.1.4, 168.24.0-28.a.1.1, 168.24.0-28.a.1.2, 168.24.0-28.a.1.3, 168.24.0-28.a.1.4, 280.24.0-28.a.1.1, 280.24.0-28.a.1.2, 280.24.0-28.a.1.3, 280.24.0-28.a.1.4, 308.24.0-28.a.1.1, 308.24.0-28.a.1.2, 308.24.0-28.a.1.3, 308.24.0-28.a.1.4
Cyclic 28-isogeny field degree: $16$
Cyclic 28-torsion field degree: $192$
Full 28-torsion field degree: $16128$

Models

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

Rational points

This modular curve has infinitely many rational points, including 556 stored non-cuspidal points.

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{3^4\cdot7^2}\cdot\frac{x^{12}(784x^{4}-252x^{2}y^{2}+81y^{4})^{3}}{y^{4}x^{16}(28x^{2}-9y^{2})^{2}}$

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
$X(2)$ $2$ $2$ $2$ $0$ $0$
28.6.0.b.1 $28$ $2$ $2$ $0$ $0$
28.6.0.e.1 $28$ $2$ $2$ $0$ $0$

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

Covering curve Level Index Degree Genus
28.24.0.a.1 $28$ $2$ $2$ $0$
28.24.0.c.1 $28$ $2$ $2$ $0$
28.96.5.c.1 $28$ $8$ $8$ $5$
28.252.16.c.1 $28$ $21$ $21$ $16$
28.336.21.c.1 $28$ $28$ $28$ $21$
56.24.0.b.1 $56$ $2$ $2$ $0$
56.24.0.f.1 $56$ $2$ $2$ $0$
84.24.0.d.1 $84$ $2$ $2$ $0$
84.24.0.f.1 $84$ $2$ $2$ $0$
84.36.2.a.1 $84$ $3$ $3$ $2$
84.48.1.a.1 $84$ $4$ $4$ $1$
140.24.0.d.1 $140$ $2$ $2$ $0$
140.24.0.f.1 $140$ $2$ $2$ $0$
140.60.4.a.1 $140$ $5$ $5$ $4$
140.72.3.a.1 $140$ $6$ $6$ $3$
140.120.7.a.1 $140$ $10$ $10$ $7$
168.24.0.i.1 $168$ $2$ $2$ $0$
168.24.0.o.1 $168$ $2$ $2$ $0$
252.324.22.a.1 $252$ $27$ $27$ $22$
280.24.0.i.1 $280$ $2$ $2$ $0$
280.24.0.o.1 $280$ $2$ $2$ $0$
308.24.0.d.1 $308$ $2$ $2$ $0$
308.24.0.e.1 $308$ $2$ $2$ $0$
308.144.9.a.1 $308$ $12$ $12$ $9$