Invariants
Level: | $40$ | $\SL_2$-level: | $8$ | Newform level: | $32$ | ||
Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
Genus: | $1 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 16 }{2}$ | ||||||
Cusps: | $16$ (none of which are rational) | Cusp widths | $4^{8}\cdot8^{8}$ | Cusp orbits | $2^{4}\cdot4^{2}$ | ||
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: | 8K1 |
Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 40.192.1.766 |
Level structure
$\GL_2(\Z/40\Z)$-generators: | $\begin{bmatrix}1&24\\4&5\end{bmatrix}$, $\begin{bmatrix}7&16\\8&21\end{bmatrix}$, $\begin{bmatrix}27&36\\38&15\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 40.96.1.y.1 for the level structure with $-I$) |
Cyclic 40-isogeny field degree: | $12$ |
Cyclic 40-torsion field degree: | $96$ |
Full 40-torsion field degree: | $3840$ |
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 z - z^{2} + w^{2} $ |
$=$ | $10 x^{2} - 3 y^{2} + 2 y z - 2 z^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ x^{4} - 6 x^{2} y^{2} - 20 x^{2} z^{2} + 4 y^{4} + 60 y^{2} z^{2} + 225 z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle \frac{2^4}{5^2}\cdot\frac{11320312500000yz^{23}-49809375000000yz^{21}w^{2}+95090625000000yz^{19}w^{4}-103241250000000yz^{17}w^{6}+70220242968750yz^{15}w^{8}-31082920312500yz^{13}w^{10}+9016934062500yz^{11}w^{12}-1686155625000yz^{9}w^{14}+195043944375yz^{7}w^{16}-12918701250yz^{5}w^{18}+421687350yz^{3}w^{20}-4688460yzw^{22}-6996337890625z^{24}+35846484375000z^{22}w^{2}-80032148437500z^{20}w^{4}+102282343750000z^{18}w^{6}-82643753906250z^{16}w^{8}+44004021093750z^{14}w^{10}-15616353593750z^{12}w^{12}+3655861687500z^{10}w^{14}-547098911250z^{8}w^{16}+49340526875z^{6}w^{18}-2409394275z^{4}w^{20}+50923410z^{2}w^{22}-226981w^{24}}{w^{8}(15421875yz^{15}-43181250yz^{13}w^{2}+48116250yz^{11}w^{4}-27142500yz^{9}w^{6}+8140650yz^{7}w^{8}-1241100yz^{5}w^{10}+82068yz^{3}w^{12}-1512yzw^{14}-9531250z^{16}+33584375z^{14}w^{2}-47669375z^{12}w^{4}+34982750z^{10}w^{6}-14135075z^{8}w^{8}+3083450z^{6}w^{10}-327862z^{4}w^{12}+13068z^{2}w^{14}-81w^{16})}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.96.1.y.1 :
$\displaystyle X$ | $=$ | $\displaystyle z$ |
$\displaystyle Y$ | $=$ | $\displaystyle x$ |
$\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{5}w$ |
Equation of the image curve:
$0$ | $=$ | $ X^{4}-6X^{2}Y^{2}+4Y^{4}-20X^{2}Z^{2}+60Y^{2}Z^{2}+225Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
---|---|---|---|---|---|---|
8.96.1-8.k.2.5 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.96.0-40.k.2.2 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.96.0-40.k.2.3 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.96.0-40.l.2.6 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.96.0-40.l.2.13 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.96.0-40.w.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.96.0-40.w.1.15 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.96.0-40.x.1.1 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.96.0-40.x.1.13 | $40$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
40.96.1-8.k.2.4 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.96.1-40.p.1.1 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.96.1-40.p.1.4 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.96.1-40.q.2.9 | $40$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
40.96.1-40.q.2.13 | $40$ | $2$ | $2$ | $1$ | $0$ | 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 |
---|---|---|---|---|---|---|
40.960.33-40.cx.2.3 | $40$ | $5$ | $5$ | $33$ | $3$ | $1^{14}\cdot2^{9}$ |
40.1152.33-40.jz.2.3 | $40$ | $6$ | $6$ | $33$ | $3$ | $1^{14}\cdot2\cdot4^{4}$ |
40.1920.65-40.nt.1.1 | $40$ | $10$ | $10$ | $65$ | $5$ | $1^{28}\cdot2^{10}\cdot4^{4}$ |
80.384.5-80.g.1.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.384.5-80.m.1.5 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.384.5-80.bi.2.4 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.384.5-80.bk.1.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.384.5-80.ep.1.2 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.384.5-80.er.1.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.384.5-80.fn.1.5 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.384.5-80.ft.1.1 | $80$ | $2$ | $2$ | $5$ | $?$ | not computed |
80.384.9-80.jp.2.9 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
80.384.9-80.jq.2.9 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
80.384.9-80.jv.2.9 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
80.384.9-80.jw.2.9 | $80$ | $2$ | $2$ | $9$ | $?$ | not computed |
240.384.5-240.be.2.9 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.384.5-240.bk.1.13 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.384.5-240.du.1.8 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.384.5-240.dw.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.384.5-240.of.1.4 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.384.5-240.oh.2.16 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.384.5-240.qr.1.13 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.384.5-240.qx.2.13 | $240$ | $2$ | $2$ | $5$ | $?$ | not computed |
240.384.9-240.bjj.2.6 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
240.384.9-240.bjk.2.6 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
240.384.9-240.bjt.2.6 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |
240.384.9-240.bju.2.6 | $240$ | $2$ | $2$ | $9$ | $?$ | not computed |