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Properties

Label 40.48.1.gi.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 $4^{2}$
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: 40.48.1.329

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}3&8\\1&13\end{bmatrix}$, $\begin{bmatrix}15&2\\38&25\end{bmatrix}$, $\begin{bmatrix}19&38\\23&9\end{bmatrix}$, $\begin{bmatrix}21&0\\4&29\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.96.1-40.gi.1.1, 40.96.1-40.gi.1.2, 80.96.1-40.gi.1.1, 80.96.1-40.gi.1.2, 80.96.1-40.gi.1.3, 80.96.1-40.gi.1.4, 120.96.1-40.gi.1.1, 120.96.1-40.gi.1.2, 240.96.1-40.gi.1.1, 240.96.1-40.gi.1.2, 240.96.1-40.gi.1.3, 240.96.1-40.gi.1.4, 280.96.1-40.gi.1.1, 280.96.1-40.gi.1.2
Cyclic 40-isogeny field degree: $24$
Cyclic 40-torsion field degree: $384$
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 $ $=$ $ 5 x^{2} + y^{2} + y z - z^{2} $
$=$ $5 x^{2} - 2 y^{2} - 7 y z + 7 z^{2} - 2 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ x^{4} + 14 x^{2} y^{2} - 30 x^{2} z^{2} + 9 y^{4} - 60 y^{2} z^{2} + 100 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:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 2^4\cdot3^3\,\frac{442442000000yz^{11}-360165000000yz^{9}w^{2}+113705640000yz^{7}w^{4}-17223300000yz^{5}w^{6}+1230568200yz^{3}w^{8}-32148900yzw^{10}-342827000000z^{12}+349031000000z^{10}w^{2}-142429770000z^{8}w^{4}+29386980000z^{6}w^{6}-3145929300z^{4}w^{8}+155641500z^{2}w^{10}-2250423w^{12}}{110610500000yz^{11}-45214500000yz^{9}w^{2}+6483780000yz^{7}w^{4}-382941000yz^{5}w^{6}+8355150yz^{3}w^{8}-43740yzw^{10}-85706750000z^{12}+52523600000z^{10}w^{2}-11517165000z^{8}w^{4}+1102995000z^{6}w^{6}-44075475z^{4}w^{8}+566190z^{2}w^{10}-972w^{12}}$

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.ba.1 $8$ $2$ $2$ $1$ $0$ dimension zero
40.24.0.cm.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.ct.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.do.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.ee.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.24.1.z.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.km.1 $40$ $5$ $5$ $17$ $6$ $1^{14}\cdot2$
40.288.17.bbg.1 $40$ $6$ $6$ $17$ $5$ $1^{14}\cdot2$
40.480.33.bro.1 $40$ $10$ $10$ $33$ $13$ $1^{28}\cdot2^{2}$
80.96.3.ob.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.od.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.rn.1 $80$ $2$ $2$ $3$ $?$ not computed
80.96.3.rp.1 $80$ $2$ $2$ $3$ $?$ not computed
120.144.9.ffa.1 $120$ $3$ $3$ $9$ $?$ not computed
120.192.9.bsw.1 $120$ $4$ $4$ $9$ $?$ not computed
240.96.3.bsf.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bsh.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.bud.1 $240$ $2$ $2$ $3$ $?$ not computed
240.96.3.buf.1 $240$ $2$ $2$ $3$ $?$ not computed