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Properties

Label 40.60.4.w.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 $\quad(D =$ $-16$)

Other labels

Cummins and Pauli (CP) label: 20A4
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.60.4.7

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}1&28\\4&33\end{bmatrix}$, $\begin{bmatrix}5&11\\36&37\end{bmatrix}$, $\begin{bmatrix}11&30\\20&11\end{bmatrix}$, $\begin{bmatrix}21&26\\36&29\end{bmatrix}$, $\begin{bmatrix}31&30\\0&1\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.120.4-40.w.1.1, 40.120.4-40.w.1.2, 40.120.4-40.w.1.3, 40.120.4-40.w.1.4, 40.120.4-40.w.1.5, 40.120.4-40.w.1.6, 40.120.4-40.w.1.7, 40.120.4-40.w.1.8, 120.120.4-40.w.1.1, 120.120.4-40.w.1.2, 120.120.4-40.w.1.3, 120.120.4-40.w.1.4, 120.120.4-40.w.1.5, 120.120.4-40.w.1.6, 120.120.4-40.w.1.7, 120.120.4-40.w.1.8, 280.120.4-40.w.1.1, 280.120.4-40.w.1.2, 280.120.4-40.w.1.3, 280.120.4-40.w.1.4, 280.120.4-40.w.1.5, 280.120.4-40.w.1.6, 280.120.4-40.w.1.7, 280.120.4-40.w.1.8
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $192$
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 $ $=$ $ 14 x^{2} + 2 y^{2} + z^{2} - w^{2} $
$=$ $2 x^{3} - 2 x y^{2} - x z^{2} + y z w$
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Singular plane model Singular plane model

$ 0 $ $=$ $ - x^{6} + 2 x^{4} z^{2} - 7 x^{2} y^{2} z^{2} - x^{2} z^{4} - 8 y^{4} z^{2} + 8 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
$(0:0:-1:1)$, $(0:0:1:1)$

Maps between models of this curve

Birational map from canonical model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{4}z$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{4}w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -2^3\,\frac{59592960xyz^{7}w-1289153278xyz^{5}w^{3}+1613297140xyz^{3}w^{5}-227517206xyzw^{7}+2211840y^{2}z^{8}-242030960y^{2}z^{6}w^{2}+921589750y^{2}z^{4}w^{4}-450084644y^{2}z^{2}w^{6}+33482558y^{2}w^{8}+1009152z^{10}-65496440z^{8}w^{2}+182910399z^{6}w^{4}-148461757z^{4}w^{6}+32329237z^{2}w^{8}-2061215w^{10}}{5040xyz^{7}w-2716xyz^{5}w^{3}-280xyz^{3}w^{5}+196xyzw^{7}-640y^{2}z^{8}+1650y^{2}z^{6}w^{2}-450y^{2}z^{4}w^{4}+14y^{2}z^{2}w^{6}+2y^{2}w^{8}+128z^{10}+200z^{8}w^{2}-707z^{6}w^{4}+477z^{4}w^{6}-97z^{2}w^{8}-w^{10}}$

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.k.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.k.1 $8$ $5$ $5$ $0$ $0$ full Jacobian
20.30.2.c.1 $20$ $2$ $2$ $2$ $0$ $1^{2}$
40.30.2.b.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.bk.1 $40$ $2$ $2$ $8$ $2$ $1^{4}$
40.120.8.bq.1 $40$ $2$ $2$ $8$ $4$ $1^{4}$
40.120.8.cc.1 $40$ $2$ $2$ $8$ $3$ $1^{4}$
40.120.8.cg.1 $40$ $2$ $2$ $8$ $5$ $1^{4}$
40.180.10.bu.1 $40$ $3$ $3$ $10$ $5$ $1^{6}$
40.240.13.hk.1 $40$ $4$ $4$ $13$ $6$ $1^{9}$
120.120.8.cb.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.ci.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.cu.1 $120$ $2$ $2$ $8$ $?$ not computed
120.120.8.cy.1 $120$ $2$ $2$ $8$ $?$ not computed
120.180.14.dq.1 $120$ $3$ $3$ $14$ $?$ not computed
120.240.17.qw.1 $120$ $4$ $4$ $17$ $?$ not computed
280.120.8.cc.1 $280$ $2$ $2$ $8$ $?$ not computed
280.120.8.cg.1 $280$ $2$ $2$ $8$ $?$ not computed
280.120.8.cs.1 $280$ $2$ $2$ $8$ $?$ not computed
280.120.8.cw.1 $280$ $2$ $2$ $8$ $?$ not computed