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Properties

Label 20.12.0.h.1
Level $20$
Index $12$
Genus $0$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $20$ $\SL_2$-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: 20.12.0.8

Level structure

$\GL_2(\Z/20\Z)$-generators: $\begin{bmatrix}5&12\\6&7\end{bmatrix}$, $\begin{bmatrix}5&12\\13&15\end{bmatrix}$, $\begin{bmatrix}7&0\\2&1\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 20.24.0-20.h.1.1, 20.24.0-20.h.1.2, 40.24.0-20.h.1.1, 40.24.0-20.h.1.2, 40.24.0-20.h.1.3, 40.24.0-20.h.1.4, 40.24.0-20.h.1.5, 40.24.0-20.h.1.6, 60.24.0-20.h.1.1, 60.24.0-20.h.1.2, 120.24.0-20.h.1.1, 120.24.0-20.h.1.2, 120.24.0-20.h.1.3, 120.24.0-20.h.1.4, 120.24.0-20.h.1.5, 120.24.0-20.h.1.6, 140.24.0-20.h.1.1, 140.24.0-20.h.1.2, 220.24.0-20.h.1.1, 220.24.0-20.h.1.2, 260.24.0-20.h.1.1, 260.24.0-20.h.1.2, 280.24.0-20.h.1.1, 280.24.0-20.h.1.2, 280.24.0-20.h.1.3, 280.24.0-20.h.1.4, 280.24.0-20.h.1.5, 280.24.0-20.h.1.6
Cyclic 20-isogeny field degree: $6$
Cyclic 20-torsion field degree: $48$
Full 20-torsion field degree: $3840$

Models

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

Rational points

This modular curve has infinitely many rational points, including 480 stored non-cuspidal points.

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{2^6}{5}\cdot\frac{x^{12}(11x^{4}+192x^{3}y-4736x^{2}y^{2}+12288xy^{3}+45056y^{4})^{3}}{x^{12}(x-8y)^{2}(x+8y)^{2}(3x^{2}-32xy+192y^{2})^{4}}$

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
$X_0(4)$ $4$ $2$ $2$ $0$ $0$
20.6.0.a.1 $20$ $2$ $2$ $0$ $0$
20.6.0.e.1 $20$ $2$ $2$ $0$ $0$

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

Covering curve Level Index Degree Genus
20.60.4.l.1 $20$ $5$ $5$ $4$
20.72.3.p.1 $20$ $6$ $6$ $3$
20.120.7.t.1 $20$ $10$ $10$ $7$
40.24.0.bm.1 $40$ $2$ $2$ $0$
40.24.0.bn.1 $40$ $2$ $2$ $0$
40.24.0.bw.1 $40$ $2$ $2$ $0$
40.24.0.bx.1 $40$ $2$ $2$ $0$
60.36.2.t.1 $60$ $3$ $3$ $2$
60.48.1.l.1 $60$ $4$ $4$ $1$
120.24.0.cq.1 $120$ $2$ $2$ $0$
120.24.0.cr.1 $120$ $2$ $2$ $0$
120.24.0.cy.1 $120$ $2$ $2$ $0$
120.24.0.cz.1 $120$ $2$ $2$ $0$
140.96.5.l.1 $140$ $8$ $8$ $5$
140.252.16.t.1 $140$ $21$ $21$ $16$
140.336.21.t.1 $140$ $28$ $28$ $21$
180.324.22.bb.1 $180$ $27$ $27$ $22$
220.144.9.l.1 $220$ $12$ $12$ $9$
260.168.11.p.1 $260$ $14$ $14$ $11$
280.24.0.ce.1 $280$ $2$ $2$ $0$
280.24.0.cf.1 $280$ $2$ $2$ $0$
280.24.0.ci.1 $280$ $2$ $2$ $0$
280.24.0.cj.1 $280$ $2$ $2$ $0$