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Properties

Label 40.48.1.u.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 $2^{4}$
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: 40.48.1.292

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}21&32\\4&37\end{bmatrix}$, $\begin{bmatrix}23&32\\6&17\end{bmatrix}$, $\begin{bmatrix}29&28\\2&21\end{bmatrix}$, $\begin{bmatrix}35&28\\6&19\end{bmatrix}$, $\begin{bmatrix}37&32\\8&39\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.96.1-40.u.1.1, 40.96.1-40.u.1.2, 40.96.1-40.u.1.3, 40.96.1-40.u.1.4, 40.96.1-40.u.1.5, 40.96.1-40.u.1.6, 40.96.1-40.u.1.7, 40.96.1-40.u.1.8, 40.96.1-40.u.1.9, 40.96.1-40.u.1.10, 40.96.1-40.u.1.11, 40.96.1-40.u.1.12, 40.96.1-40.u.1.13, 40.96.1-40.u.1.14, 40.96.1-40.u.1.15, 40.96.1-40.u.1.16, 120.96.1-40.u.1.1, 120.96.1-40.u.1.2, 120.96.1-40.u.1.3, 120.96.1-40.u.1.4, 120.96.1-40.u.1.5, 120.96.1-40.u.1.6, 120.96.1-40.u.1.7, 120.96.1-40.u.1.8, 120.96.1-40.u.1.9, 120.96.1-40.u.1.10, 120.96.1-40.u.1.11, 120.96.1-40.u.1.12, 120.96.1-40.u.1.13, 120.96.1-40.u.1.14, 120.96.1-40.u.1.15, 120.96.1-40.u.1.16, 280.96.1-40.u.1.1, 280.96.1-40.u.1.2, 280.96.1-40.u.1.3, 280.96.1-40.u.1.4, 280.96.1-40.u.1.5, 280.96.1-40.u.1.6, 280.96.1-40.u.1.7, 280.96.1-40.u.1.8, 280.96.1-40.u.1.9, 280.96.1-40.u.1.10, 280.96.1-40.u.1.11, 280.96.1-40.u.1.12, 280.96.1-40.u.1.13, 280.96.1-40.u.1.14, 280.96.1-40.u.1.15, 280.96.1-40.u.1.16
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $192$
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 $ $=$ $ 3 x^{2} - x y + 2 x z - 2 y^{2} - 2 y z + 2 z^{2} $
$=$ $10 x y + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 25 x^{4} + 5 x^{3} y - x^{2} y^{2} - 5 x^{2} z^{2} + 2 x y z^{2} - 6 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle 5z$
$\displaystyle Z$ $=$ $\displaystyle \frac{1}{2}w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{2^4}{3^8\cdot5}\cdot\frac{630784000000xz^{11}-46802136000000xz^{9}w^{2}-11618346240000xz^{7}w^{4}-1134083484000xz^{5}w^{6}-45930823200xz^{3}w^{8}-652695570xzw^{10}-504561760000000y^{2}z^{10}-140563080000000y^{2}z^{8}w^{2}-16274048400000y^{2}z^{6}w^{4}-856302300000y^{2}z^{4}w^{6}-19177560000y^{2}z^{2}w^{8}-87637950y^{2}w^{10}+311731616000000yz^{11}+111752616000000yz^{9}w^{2}+17400223440000yz^{7}w^{4}+1339110684000yz^{5}w^{6}+48681826200yz^{3}w^{8}+652695570yzw^{10}+47104000000z^{12}-30854492800000z^{10}w^{2}-10644621120000z^{8}w^{4}-1381610880000z^{6}w^{6}-80232130800z^{4}w^{8}-1882700820z^{2}w^{10}-14432499w^{12}}{w^{8}(640xz^{3}+138xzw^{2}+2000y^{2}z^{2}+30y^{2}w^{2}-1240yz^{3}-138yzw^{2}+160z^{4}+188z^{2}w^{2}+3w^{4})}$

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.d.1 $8$ $2$ $2$ $1$ $0$ dimension zero
20.24.0.c.1 $20$ $2$ $2$ $0$ $0$ full Jacobian
40.24.0.l.1 $40$ $2$ $2$ $0$ $0$ full Jacobian

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
40.96.1.ba.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.ba.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.bc.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.bc.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.be.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.be.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.bg.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1.bg.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.240.17.bh.1 $40$ $5$ $5$ $17$ $5$ $1^{14}\cdot2$
40.288.17.ck.1 $40$ $6$ $6$ $17$ $3$ $1^{14}\cdot2$
40.480.33.fr.1 $40$ $10$ $10$ $33$ $9$ $1^{28}\cdot2^{2}$
120.96.1.dk.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.dk.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.dm.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.dm.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.do.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.do.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.dq.1 $120$ $2$ $2$ $1$ $?$ dimension zero
120.96.1.dq.2 $120$ $2$ $2$ $1$ $?$ dimension zero
120.144.9.dx.1 $120$ $3$ $3$ $9$ $?$ not computed
120.192.9.ci.1 $120$ $4$ $4$ $9$ $?$ not computed
280.96.1.dc.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.dc.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.de.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.de.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.dg.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.dg.2 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.di.1 $280$ $2$ $2$ $1$ $?$ dimension zero
280.96.1.di.2 $280$ $2$ $2$ $1$ $?$ dimension zero