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Properties

Label 12.12.0.h.1
Level $12$
Index $12$
Genus $0$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $12$ $\SL_2$-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: 12.12.0.11

Level structure

$\GL_2(\Z/12\Z)$-generators: $\begin{bmatrix}1&11\\4&3\end{bmatrix}$, $\begin{bmatrix}9&8\\8&7\end{bmatrix}$, $\begin{bmatrix}11&0\\4&11\end{bmatrix}$
$\GL_2(\Z/12\Z)$-subgroup: $C_2^3:\GL(2,3)$
Contains $-I$: yes
Quadratic refinements: 12.24.0-12.h.1.1, 12.24.0-12.h.1.2, 24.24.0-12.h.1.1, 24.24.0-12.h.1.2, 24.24.0-12.h.1.3, 24.24.0-12.h.1.4, 24.24.0-12.h.1.5, 24.24.0-12.h.1.6, 60.24.0-12.h.1.1, 60.24.0-12.h.1.2, 84.24.0-12.h.1.1, 84.24.0-12.h.1.2, 120.24.0-12.h.1.1, 120.24.0-12.h.1.2, 120.24.0-12.h.1.3, 120.24.0-12.h.1.4, 120.24.0-12.h.1.5, 120.24.0-12.h.1.6, 132.24.0-12.h.1.1, 132.24.0-12.h.1.2, 156.24.0-12.h.1.1, 156.24.0-12.h.1.2, 168.24.0-12.h.1.1, 168.24.0-12.h.1.2, 168.24.0-12.h.1.3, 168.24.0-12.h.1.4, 168.24.0-12.h.1.5, 168.24.0-12.h.1.6, 204.24.0-12.h.1.1, 204.24.0-12.h.1.2, 228.24.0-12.h.1.1, 228.24.0-12.h.1.2, 264.24.0-12.h.1.1, 264.24.0-12.h.1.2, 264.24.0-12.h.1.3, 264.24.0-12.h.1.4, 264.24.0-12.h.1.5, 264.24.0-12.h.1.6, 276.24.0-12.h.1.1, 276.24.0-12.h.1.2, 312.24.0-12.h.1.1, 312.24.0-12.h.1.2, 312.24.0-12.h.1.3, 312.24.0-12.h.1.4, 312.24.0-12.h.1.5, 312.24.0-12.h.1.6
Cyclic 12-isogeny field degree: $4$
Cyclic 12-torsion field degree: $16$
Full 12-torsion field degree: $384$

Models

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

Rational points

This modular curve has infinitely many rational points, including 771 stored non-cuspidal points.

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{3}\cdot\frac{(3x-2y)^{12}(9x^{4}+42x^{2}y^{2}+y^{4})^{3}}{y^{2}x^{2}(3x-2y)^{12}(3x^{2}-y^{2})^{4}}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
$X_0(4)$ $4$ $2$ $2$ $0$ $0$
12.6.0.a.1 $12$ $2$ $2$ $0$ $0$
12.6.0.g.1 $12$ $2$ $2$ $0$ $0$

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

Covering curve Level Index Degree Genus
12.36.2.t.1 $12$ $3$ $3$ $2$
12.48.1.l.1 $12$ $4$ $4$ $1$
24.24.0.bk.1 $24$ $2$ $2$ $0$
24.24.0.bl.1 $24$ $2$ $2$ $0$
24.24.0.bs.1 $24$ $2$ $2$ $0$
24.24.0.bt.1 $24$ $2$ $2$ $0$
36.324.22.bb.1 $36$ $27$ $27$ $22$
60.60.4.l.1 $60$ $5$ $5$ $4$
60.72.3.et.1 $60$ $6$ $6$ $3$
60.120.7.t.1 $60$ $10$ $10$ $7$
84.96.5.l.1 $84$ $8$ $8$ $5$
84.252.16.t.1 $84$ $21$ $21$ $16$
84.336.21.t.1 $84$ $28$ $28$ $21$
120.24.0.cc.1 $120$ $2$ $2$ $0$
120.24.0.cd.1 $120$ $2$ $2$ $0$
120.24.0.cg.1 $120$ $2$ $2$ $0$
120.24.0.ch.1 $120$ $2$ $2$ $0$
132.144.9.l.1 $132$ $12$ $12$ $9$
156.168.11.p.1 $156$ $14$ $14$ $11$
168.24.0.by.1 $168$ $2$ $2$ $0$
168.24.0.bz.1 $168$ $2$ $2$ $0$
168.24.0.cc.1 $168$ $2$ $2$ $0$
168.24.0.cd.1 $168$ $2$ $2$ $0$
204.216.15.p.1 $204$ $18$ $18$ $15$
228.240.17.l.1 $228$ $20$ $20$ $17$
264.24.0.by.1 $264$ $2$ $2$ $0$
264.24.0.bz.1 $264$ $2$ $2$ $0$
264.24.0.cc.1 $264$ $2$ $2$ $0$
264.24.0.cd.1 $264$ $2$ $2$ $0$
276.288.21.l.1 $276$ $24$ $24$ $21$
312.24.0.cc.1 $312$ $2$ $2$ $0$
312.24.0.cd.1 $312$ $2$ $2$ $0$
312.24.0.cg.1 $312$ $2$ $2$ $0$
312.24.0.ch.1 $312$ $2$ $2$ $0$