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Properties

Label 40.48.1.fc.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: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8F1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.48.1.417

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}1&36\\33&11\end{bmatrix}$, $\begin{bmatrix}5&14\\36&3\end{bmatrix}$, $\begin{bmatrix}25&18\\18&15\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 80.96.1-40.fc.1.1, 80.96.1-40.fc.1.2, 80.96.1-40.fc.1.3, 80.96.1-40.fc.1.4, 80.96.1-40.fc.1.5, 80.96.1-40.fc.1.6, 240.96.1-40.fc.1.1, 240.96.1-40.fc.1.2, 240.96.1-40.fc.1.3, 240.96.1-40.fc.1.4, 240.96.1-40.fc.1.5, 240.96.1-40.fc.1.6
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 $ $=$ $ y^{2} - y w - z^{2} - w^{2} $
$=$ $10 x^{2} - 3 y^{2} - 2 y w - 2 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 225 x^{4} - 60 x^{2} y^{2} + 20 x^{2} z^{2} + 4 y^{4} - 6 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:

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle 5x$
$\displaystyle Z$ $=$ $\displaystyle 5w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 2^8\,\frac{5760yz^{10}w+142200yz^{8}w^{3}+1055250yz^{6}w^{5}+3279375yz^{4}w^{7}+4500000yz^{2}w^{9}+2250000yw^{11}+512z^{12}+32160z^{10}w^{2}+387300z^{8}w^{4}+1796875z^{6}w^{6}+3838125z^{4}w^{8}+3787500z^{2}w^{10}+1390625w^{12}}{z^{8}(30yz^{2}w+75yw^{3}+9z^{4}+55z^{2}w^{2}+50w^{4})}$

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
8.24.1.u.1 $8$ $2$ $2$ $1$ $0$ dimension zero
40.24.0.cc.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.ck.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.de.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.dw.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.1.ba.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.24.1.bu.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.jg.1 $40$ $5$ $5$ $17$ $5$ $1^{14}\cdot2$
40.288.17.xm.1 $40$ $6$ $6$ $17$ $7$ $1^{14}\cdot2$
40.480.33.bpa.1 $40$ $10$ $10$ $33$ $12$ $1^{28}\cdot2^{2}$
80.96.3.pn.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.po.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.pp.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.pq.1 $80$ $2$ $2$ $3$ $?$ not computed
120.144.9.ezg.1 $120$ $3$ $3$ $9$ $?$ not computed
120.192.9.bpc.1 $120$ $4$ $4$ $9$ $?$ not computed
240.96.3.bqx.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bqy.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bqz.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bra.1 $240$ $2$ $2$ $3$ $?$ not computed