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Properties

Label 40.48.1.iv.2
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 $2^{2}\cdot4$
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: 8G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.48.1.220

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}15&34\\31&5\end{bmatrix}$, $\begin{bmatrix}25&28\\14&13\end{bmatrix}$, $\begin{bmatrix}39&12\\8&25\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 80.96.1-40.iv.2.1, 80.96.1-40.iv.2.2, 80.96.1-40.iv.2.3, 80.96.1-40.iv.2.4, 80.96.1-40.iv.2.5, 80.96.1-40.iv.2.6, 80.96.1-40.iv.2.7, 80.96.1-40.iv.2.8, 80.96.1-40.iv.2.9, 80.96.1-40.iv.2.10, 80.96.1-40.iv.2.11, 80.96.1-40.iv.2.12, 80.96.1-40.iv.2.13, 80.96.1-40.iv.2.14, 80.96.1-40.iv.2.15, 80.96.1-40.iv.2.16, 240.96.1-40.iv.2.1, 240.96.1-40.iv.2.2, 240.96.1-40.iv.2.3, 240.96.1-40.iv.2.4, 240.96.1-40.iv.2.5, 240.96.1-40.iv.2.6, 240.96.1-40.iv.2.7, 240.96.1-40.iv.2.8, 240.96.1-40.iv.2.9, 240.96.1-40.iv.2.10, 240.96.1-40.iv.2.11, 240.96.1-40.iv.2.12, 240.96.1-40.iv.2.13, 240.96.1-40.iv.2.14, 240.96.1-40.iv.2.15, 240.96.1-40.iv.2.16
Cyclic 40-isogeny field degree: $24$
Cyclic 40-torsion field degree: $384$
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 $ $=$ $ x^{2} + 3 x y + x z + y^{2} - y z - 2 z^{2} $
$=$ $2 x^{2} + x y + 5 x z + 2 y^{2} - 5 y z + 6 z^{2} - 2 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 52 x^{4} - 40 x^{3} y + 160 x^{2} y^{2} - 34 x^{2} z^{2} - 100 x y^{3} + 20 x y z^{2} + 25 y^{4} + \cdots + 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:

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle y$
$\displaystyle Z$ $=$ $\displaystyle w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 2^4\,\frac{1728xz^{9}w^{2}-2304xz^{7}w^{4}+224xz^{5}w^{6}+320xz^{3}w^{8}+12xzw^{10}-1728yz^{9}w^{2}+2304yz^{7}w^{4}-224yz^{5}w^{6}-320yz^{3}w^{8}-12yzw^{10}+1728z^{12}-5040z^{8}w^{4}+3200z^{6}w^{6}-92z^{4}w^{8}-96z^{2}w^{10}-w^{12}}{z^{8}(4xzw^{2}-4yzw^{2}+16z^{4}-w^{4})}$

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 Kernel decomposition
8.24.0.bl.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.el.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.1.ef.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.yr.2 $40$ $5$ $5$ $17$ $3$ $1^{6}\cdot2^{5}$
40.288.17.cgt.2 $40$ $6$ $6$ $17$ $3$ $1^{6}\cdot2\cdot4^{2}$
40.480.33.dxz.2 $40$ $10$ $10$ $33$ $4$ $1^{12}\cdot2^{6}\cdot4^{2}$
80.96.5.hy.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.ia.1 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.ju.1 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.jw.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.ta.1 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.tc.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.uw.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.uy.1 $80$ $2$ $2$ $5$ $?$ not computed
120.144.9.bezd.2 $120$ $3$ $3$ $9$ $?$ not computed
120.192.9.dpg.1 $120$ $4$ $4$ $9$ $?$ not computed
240.96.5.cga.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.cgc.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.cgy.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.cha.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.drk.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.drm.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.dsi.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.dsk.1 $240$ $2$ $2$ $5$ $?$ not computed