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Properties

Label 40.48.1.bn.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: $1600$
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 $2^{4}$
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: 40.48.1.117

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}7&32\\6&15\end{bmatrix}$, $\begin{bmatrix}9&16\\36&21\end{bmatrix}$, $\begin{bmatrix}27&2\\8&21\end{bmatrix}$, $\begin{bmatrix}33&22\\12&19\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.96.1-40.bn.1.1, 40.96.1-40.bn.1.2, 40.96.1-40.bn.1.3, 40.96.1-40.bn.1.4, 80.96.1-40.bn.1.1, 80.96.1-40.bn.1.2, 80.96.1-40.bn.1.3, 80.96.1-40.bn.1.4, 120.96.1-40.bn.1.1, 120.96.1-40.bn.1.2, 120.96.1-40.bn.1.3, 120.96.1-40.bn.1.4, 240.96.1-40.bn.1.1, 240.96.1-40.bn.1.2, 240.96.1-40.bn.1.3, 240.96.1-40.bn.1.4, 280.96.1-40.bn.1.1, 280.96.1-40.bn.1.2, 280.96.1-40.bn.1.3, 280.96.1-40.bn.1.4
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $192$
Full 40-torsion field degree: $15360$

Jacobian

Conductor: $2^{6}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 1600.2.a.n

Models

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

$ 0 $ $=$ $ 2 x y - z^{2} $
$=$ $10 x^{2} + 10 y^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 10 x^{2} y^{2} + 4 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{10}w$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}z$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{2^8}{5^4}\cdot\frac{(5z^{2}-w^{2})^{3}(5z^{2}+w^{2})^{3}}{w^{4}z^{8}}$

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
8.24.0.g.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.a.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.dk.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.eg.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.1.b.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.24.1.cw.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.24.1.ds.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.cl.1 $40$ $5$ $5$ $17$ $4$ $1^{14}\cdot2$
40.288.17.fp.1 $40$ $6$ $6$ $17$ $6$ $1^{14}\cdot2$
40.480.33.lp.1 $40$ $10$ $10$ $33$ $8$ $1^{28}\cdot2^{2}$
80.96.3.cy.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.cz.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.da.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.db.1 $80$ $2$ $2$ $3$ $?$ not computed
120.144.9.bbf.1 $120$ $3$ $3$ $9$ $?$ not computed
120.192.9.ob.1 $120$ $4$ $4$ $9$ $?$ not computed
240.96.3.ho.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.hp.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.hq.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.hr.1 $240$ $2$ $2$ $3$ $?$ not computed