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Properties

Label 40.60.4.a.1
Level $40$
Index $60$
Genus $4$
Analytic rank $2$
Cusps $4$
$\Q$-cusps $2$

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Invariants

Level: $40$ $\SL_2$-level: $20$ Newform level: $1600$
Index: $60$ $\PSL_2$-index:$60$
Genus: $4 = 1 + \frac{ 60 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (of which $2$ are rational) Cusp widths $10^{2}\cdot20^{2}$ Cusp orbits $1^{2}\cdot2$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $2$
$\Q$-gonality: $3 \le \gamma \le 4$
$\overline{\Q}$-gonality: $3 \le \gamma \le 4$
Rational cusps: $2$
Rational CM points: yes $\quad(D =$ $-4$)

Other labels

Cummins and Pauli (CP) label: 20A4
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.60.4.1

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}1&14\\22&37\end{bmatrix}$, $\begin{bmatrix}5&38\\4&19\end{bmatrix}$, $\begin{bmatrix}11&0\\8&19\end{bmatrix}$, $\begin{bmatrix}21&14\\6&35\end{bmatrix}$, $\begin{bmatrix}33&38\\18&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.120.4-40.a.1.1, 40.120.4-40.a.1.2, 40.120.4-40.a.1.3, 40.120.4-40.a.1.4, 40.120.4-40.a.1.5, 40.120.4-40.a.1.6, 40.120.4-40.a.1.7, 40.120.4-40.a.1.8, 120.120.4-40.a.1.1, 120.120.4-40.a.1.2, 120.120.4-40.a.1.3, 120.120.4-40.a.1.4, 120.120.4-40.a.1.5, 120.120.4-40.a.1.6, 120.120.4-40.a.1.7, 120.120.4-40.a.1.8, 280.120.4-40.a.1.1, 280.120.4-40.a.1.2, 280.120.4-40.a.1.3, 280.120.4-40.a.1.4, 280.120.4-40.a.1.5, 280.120.4-40.a.1.6, 280.120.4-40.a.1.7, 280.120.4-40.a.1.8
Cyclic 40-isogeny field degree: $24$
Cyclic 40-torsion field degree: $384$
Full 40-torsion field degree: $12288$

Jacobian

Conductor: $2^{14}\cdot5^{8}$
Simple: no
Squarefree: no
Decomposition: $1^{4}$
Newforms: 50.2.a.b$^{2}$, 1600.2.a.a, 1600.2.a.q

Models

Canonical model in $\mathbb{P}^{ 3 }$

$ 0 $ $=$ $ 2 x^{2} - 2 x y + 4 y^{2} - z w - w^{2} $
$=$ $2 x^{3} + 2 x^{2} y - x z^{2} - 3 x z w - x w^{2} + y z^{2} + y z w$
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Singular plane model Singular plane model

$ 0 $ $=$ $ - 2 x^{6} - 5 x^{4} y^{2} - x^{4} y z + 6 x^{4} z^{2} - 4 x^{2} y^{4} - 6 x^{2} y^{3} z + \cdots + 4 y^{2} z^{4} $
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Rational points

This modular curve has 2 rational cusps and 1 rational CM point, but no other known rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:0:-1:1)$, $(0:0:1:0)$

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x-y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{2}z$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}w$

Maps to other modular curves

$j$-invariant map of degree 60 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^4\,\frac{4502xyz^{8}+8646xyz^{7}w-27880xyz^{6}w^{2}-75388xyz^{5}w^{3}+28730xyz^{4}w^{4}+239356xyz^{3}w^{5}+227864xyz^{2}w^{6}+65104xyzw^{7}+2286y^{2}z^{8}+15382y^{2}z^{7}w+6608y^{2}z^{6}w^{2}-172044y^{2}z^{5}w^{3}-443290y^{2}z^{4}w^{4}-326254y^{2}z^{3}w^{5}+121028y^{2}z^{2}w^{6}+227864y^{2}zw^{7}+65104y^{2}w^{8}-1024z^{10}-6817z^{9}w-18947z^{8}w^{2}-18464z^{7}w^{3}+42554z^{6}w^{4}+150188z^{5}w^{5}+147652z^{4}w^{6}-14656z^{3}w^{7}-126544z^{2}w^{8}-81920zw^{9}-16384w^{10}}{14xyz^{8}+50xyz^{7}w+82xyz^{6}w^{2}+106xyz^{5}w^{3}+70xyz^{4}w^{4}+14xyz^{3}w^{5}-14xyz^{2}w^{6}-4xyzw^{7}+6y^{2}z^{8}+50y^{2}z^{7}w+66y^{2}z^{6}w^{2}+10y^{2}z^{5}w^{3}-10y^{2}z^{4}w^{4}-26y^{2}z^{3}w^{5}-18y^{2}z^{2}w^{6}-14y^{2}zw^{7}-4y^{2}w^{8}-5z^{9}w-29z^{8}w^{2}-59z^{7}w^{3}-53z^{6}w^{4}-17z^{5}w^{5}+13z^{4}w^{6}+16z^{3}w^{7}+4z^{2}w^{8}}$

Modular covers

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Cover information

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The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_{S_4}(5)$ $5$ $12$ $12$ $0$ $0$ full Jacobian
8.12.0.a.1 $8$ $5$ $5$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.12.0.a.1 $8$ $5$ $5$ $0$ $0$ full Jacobian
10.30.2.a.1 $10$ $2$ $2$ $2$ $0$ $1^{2}$
40.30.2.c.1 $40$ $2$ $2$ $2$ $1$ $1^{2}$
40.30.2.p.1 $40$ $2$ $2$ $2$ $1$ $1^{2}$

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
40.120.8.a.1 $40$ $2$ $2$ $8$ $5$ $1^{4}$
40.120.8.c.1 $40$ $2$ $2$ $8$ $2$ $1^{4}$
40.120.8.e.1 $40$ $2$ $2$ $8$ $3$ $1^{4}$
40.120.8.g.1 $40$ $2$ $2$ $8$ $4$ $1^{4}$
40.180.10.a.1 $40$ $3$ $3$ $10$ $5$ $1^{6}$
40.240.13.y.1 $40$ $4$ $4$ $13$ $6$ $1^{9}$
120.120.8.b.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.d.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.i.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.k.1 $120$ $2$ $2$ $8$ $?$ not computed
120.180.14.a.1 $120$ $3$ $3$ $14$ $?$ not computed
120.240.17.fg.1 $120$ $4$ $4$ $17$ $?$ not computed
280.120.8.b.1 $280$ $2$ $2$ $8$ $?$ not computed
280.120.8.d.1 $280$ $2$ $2$ $8$ $?$ not computed
280.120.8.i.1 $280$ $2$ $2$ $8$ $?$ not computed
280.120.8.k.1 $280$ $2$ $2$ $8$ $?$ not computed