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Properties

Label 40.48.1.ga.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: $32$
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 $4^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2 \le \gamma \le 4$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8G1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.48.1.626

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}9&6\\7&15\end{bmatrix}$, $\begin{bmatrix}17&24\\20&3\end{bmatrix}$, $\begin{bmatrix}21&38\\21&21\end{bmatrix}$, $\begin{bmatrix}35&8\\28&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.96.1-40.ga.1.1, 40.96.1-40.ga.1.2, 40.96.1-40.ga.1.3, 40.96.1-40.ga.1.4, 40.96.1-40.ga.1.5, 40.96.1-40.ga.1.6, 40.96.1-40.ga.1.7, 40.96.1-40.ga.1.8, 80.96.1-40.ga.1.1, 80.96.1-40.ga.1.2, 80.96.1-40.ga.1.3, 80.96.1-40.ga.1.4, 80.96.1-40.ga.1.5, 80.96.1-40.ga.1.6, 80.96.1-40.ga.1.7, 80.96.1-40.ga.1.8, 120.96.1-40.ga.1.1, 120.96.1-40.ga.1.2, 120.96.1-40.ga.1.3, 120.96.1-40.ga.1.4, 120.96.1-40.ga.1.5, 120.96.1-40.ga.1.6, 120.96.1-40.ga.1.7, 120.96.1-40.ga.1.8, 240.96.1-40.ga.1.1, 240.96.1-40.ga.1.2, 240.96.1-40.ga.1.3, 240.96.1-40.ga.1.4, 240.96.1-40.ga.1.5, 240.96.1-40.ga.1.6, 240.96.1-40.ga.1.7, 240.96.1-40.ga.1.8, 280.96.1-40.ga.1.1, 280.96.1-40.ga.1.2, 280.96.1-40.ga.1.3, 280.96.1-40.ga.1.4, 280.96.1-40.ga.1.5, 280.96.1-40.ga.1.6, 280.96.1-40.ga.1.7, 280.96.1-40.ga.1.8
Cyclic 40-isogeny field degree: $24$
Cyclic 40-torsion field degree: $384$
Full 40-torsion field degree: $15360$

Jacobian

Conductor: $2^{5}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 32.2.a.a

Models

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

$ 0 $ $=$ $ 10 x y - z^{2} + z w + w^{2} $
$=$ $20 x^{2} + 2 y^{2} - z^{2} - w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} - 2 x^{3} z - 5 x^{2} y^{2} - x^{2} z^{2} + 2 x z^{3} + 10 y^{4} - 5 y^{2} z^{2} + 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 $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^4\cdot5^2\,\frac{9y^{2}z^{10}-288y^{2}z^{9}w-1107y^{2}z^{8}w^{2}-2736y^{2}z^{7}w^{3}-2106y^{2}z^{6}w^{4}-2106y^{2}z^{4}w^{6}+2736y^{2}z^{3}w^{7}-1107y^{2}z^{2}w^{8}+288y^{2}zw^{9}+9y^{2}w^{10}-7z^{12}+66z^{11}w+513z^{10}w^{2}+1726z^{9}w^{3}+2037z^{8}w^{4}+2100z^{7}w^{5}+694z^{6}w^{6}-2100z^{5}w^{7}+2037z^{4}w^{8}-1726z^{3}w^{9}+513z^{2}w^{10}-66zw^{11}-7w^{12}}{(z^{2}-zw-w^{2})^{4}(5y^{2}z^{2}+5y^{2}w^{2}-z^{4}+2z^{3}w+z^{2}w^{2}-2zw^{3}-w^{4})}$

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
40.24.1.s.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.ke.1 $40$ $5$ $5$ $17$ $1$ $1^{6}\cdot2^{3}\cdot4$
40.288.17.yu.2 $40$ $6$ $6$ $17$ $1$ $1^{6}\cdot2\cdot4^{2}$
40.480.33.bpy.1 $40$ $10$ $10$ $33$ $3$ $1^{12}\cdot2^{4}\cdot4^{3}$
80.96.5.mj.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.ml.1 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.mn.1 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.mp.1 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.mz.1 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.nb.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.nd.2 $80$ $2$ $2$ $5$ $?$ not computed
80.96.5.nf.2 $80$ $2$ $2$ $5$ $?$ not computed
120.144.9.fae.2 $120$ $3$ $3$ $9$ $?$ not computed
120.192.9.bqf.1 $120$ $4$ $4$ $9$ $?$ not computed
240.96.5.bmh.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.bmj.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.bml.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.bmn.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.bnn.2 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.bnp.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.bnr.1 $240$ $2$ $2$ $5$ $?$ not computed
240.96.5.bnt.2 $240$ $2$ $2$ $5$ $?$ not computed