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Properties

Label 40.48.1.fn.1
Level $40$
Index $48$
Genus $1$
Analytic rank $0$
Cusps $8$
$\Q$-cusps $0$

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Invariants

Level: $40$ $\SL_2$-level: $8$ Newform level: $64$
Index: $48$ $\PSL_2$-index:$48$
Genus: $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$
Cusps: $8$ (none of which are rational) Cusp widths $4^{4}\cdot8^{4}$ Cusp orbits $2^{2}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8F1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.48.1.394

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}13&2\\9&15\end{bmatrix}$, $\begin{bmatrix}25&8\\23&39\end{bmatrix}$, $\begin{bmatrix}27&8\\26&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 80.96.1-40.fn.1.1, 80.96.1-40.fn.1.2, 80.96.1-40.fn.1.3, 80.96.1-40.fn.1.4, 80.96.1-40.fn.1.5, 80.96.1-40.fn.1.6, 80.96.1-40.fn.1.7, 80.96.1-40.fn.1.8, 240.96.1-40.fn.1.1, 240.96.1-40.fn.1.2, 240.96.1-40.fn.1.3, 240.96.1-40.fn.1.4, 240.96.1-40.fn.1.5, 240.96.1-40.fn.1.6, 240.96.1-40.fn.1.7, 240.96.1-40.fn.1.8
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $192$
Full 40-torsion field degree: $15360$

Jacobian

Conductor: $2^{6}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 64.2.a.a

Models

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

$ 0 $ $=$ $ y^{2} - y z - z^{2} - w^{2} $
$=$ $80 x^{2} - 3 y^{2} - 2 y z - 2 z^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 3 x^{2} y^{2} + 20 x^{2} z^{2} + y^{4} - 30 y^{2} z^{2} + 225 z^{4} $
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Rational points

This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle 4x$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{5}w$

Maps to other modular curves

$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -2^4\,\frac{2250000yz^{11}+4500000yz^{9}w^{2}+2688750yz^{7}w^{4}+346500yz^{5}w^{6}-50175yz^{3}w^{8}+4410yzw^{10}+1390625z^{12}+3787500z^{10}w^{2}+3472500z^{8}w^{4}+1093750z^{6}w^{6}+5925z^{4}w^{8}-10815z^{2}w^{10}-343w^{12}}{w^{4}(13125yz^{7}+15750yz^{5}w^{2}+5550yz^{3}w^{4}+540yzw^{6}+8125z^{8}+15625z^{6}w^{2}+9325z^{4}w^{4}+1890z^{2}w^{6}+81w^{8})}$

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 Kernel decomposition
8.24.1.t.1 $8$ $2$ $2$ $1$ $0$ dimension zero
40.24.0.ch.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.cv.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.dk.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.dt.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.1.bc.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.24.1.bi.1 $40$ $2$ $2$ $1$ $0$ dimension zero

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

Covering curve Level Index Degree Genus Rank Kernel decomposition
40.240.17.jr.1 $40$ $5$ $5$ $17$ $2$ $1^{14}\cdot2$
40.288.17.xx.1 $40$ $6$ $6$ $17$ $5$ $1^{14}\cdot2$
40.480.33.bpl.1 $40$ $10$ $10$ $33$ $4$ $1^{28}\cdot2^{2}$
80.96.3.pz.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.qa.1 $80$ $2$ $2$ $3$ $?$ not computed
120.144.9.ezr.1 $120$ $3$ $3$ $9$ $?$ not computed
120.192.9.bpn.1 $120$ $4$ $4$ $9$ $?$ not computed
240.96.3.brj.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.brk.1 $240$ $2$ $2$ $3$ $?$ not computed