↑

Properties

Label 40.96.1.y.1
Level $40$
Index $96$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

Related objects

Downloads

Learn more

Invariants

Level: $40$ $\SL_2$-level: $8$ Newform level: $32$
Index: $96$ $\PSL_2$-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: 40.96.1.1036

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}5&36\\22&39\end{bmatrix}$, $\begin{bmatrix}19&4\\34&21\end{bmatrix}$, $\begin{bmatrix}23&32\\12&5\end{bmatrix}$, $\begin{bmatrix}39&32\\4&31\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.192.1-40.y.1.1, 40.192.1-40.y.1.2, 40.192.1-40.y.1.3, 40.192.1-40.y.1.4, 40.192.1-40.y.1.5, 40.192.1-40.y.1.6, 40.192.1-40.y.1.7, 40.192.1-40.y.1.8, 80.192.1-40.y.1.1, 80.192.1-40.y.1.2, 80.192.1-40.y.1.3, 80.192.1-40.y.1.4, 80.192.1-40.y.1.5, 80.192.1-40.y.1.6, 80.192.1-40.y.1.7, 80.192.1-40.y.1.8, 80.192.1-40.y.1.9, 80.192.1-40.y.1.10, 80.192.1-40.y.1.11, 80.192.1-40.y.1.12, 120.192.1-40.y.1.1, 120.192.1-40.y.1.2, 120.192.1-40.y.1.3, 120.192.1-40.y.1.4, 120.192.1-40.y.1.5, 120.192.1-40.y.1.6, 120.192.1-40.y.1.7, 120.192.1-40.y.1.8, 240.192.1-40.y.1.1, 240.192.1-40.y.1.2, 240.192.1-40.y.1.3, 240.192.1-40.y.1.4, 240.192.1-40.y.1.5, 240.192.1-40.y.1.6, 240.192.1-40.y.1.7, 240.192.1-40.y.1.8, 240.192.1-40.y.1.9, 240.192.1-40.y.1.10, 240.192.1-40.y.1.11, 240.192.1-40.y.1.12, 280.192.1-40.y.1.1, 280.192.1-40.y.1.2, 280.192.1-40.y.1.3, 280.192.1-40.y.1.4, 280.192.1-40.y.1.5, 280.192.1-40.y.1.6, 280.192.1-40.y.1.7, 280.192.1-40.y.1.8
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $192$
Full 40-torsion field degree: $7680$

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} $
$=$ $10 x^{2} - 3 y^{2} + 2 y z - 2 z^{2}$
 Copy content Toggle raw display

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} $
 Copy content Toggle raw display

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 96 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle \frac{2^4}{5^2}\cdot\frac{11320312500000yz^{23}-49809375000000yz^{21}w^{2}+95090625000000yz^{19}w^{4}-103241250000000yz^{17}w^{6}+70220242968750yz^{15}w^{8}-31082920312500yz^{13}w^{10}+9016934062500yz^{11}w^{12}-1686155625000yz^{9}w^{14}+195043944375yz^{7}w^{16}-12918701250yz^{5}w^{18}+421687350yz^{3}w^{20}-4688460yzw^{22}-6996337890625z^{24}+35846484375000z^{22}w^{2}-80032148437500z^{20}w^{4}+102282343750000z^{18}w^{6}-82643753906250z^{16}w^{8}+44004021093750z^{14}w^{10}-15616353593750z^{12}w^{12}+3655861687500z^{10}w^{14}-547098911250z^{8}w^{16}+49340526875z^{6}w^{18}-2409394275z^{4}w^{20}+50923410z^{2}w^{22}-226981w^{24}}{w^{8}(15421875yz^{15}-43181250yz^{13}w^{2}+48116250yz^{11}w^{4}-27142500yz^{9}w^{6}+8140650yz^{7}w^{8}-1241100yz^{5}w^{10}+82068yz^{3}w^{12}-1512yzw^{14}-9531250z^{16}+33584375z^{14}w^{2}-47669375z^{12}w^{4}+34982750z^{10}w^{6}-14135075z^{8}w^{8}+3083450z^{6}w^{10}-327862z^{4}w^{12}+13068z^{2}w^{14}-81w^{16})}$

Modular covers

Sorry, your browser does not support the nearby lattice.

Cover information

Click on a modular curve in the diagram to see information about it.
See a full page version of the diagram

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.48.1.k.2 $8$ $2$ $2$ $1$ $0$ dimension zero
40.48.0.k.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.l.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.w.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.x.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.1.p.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.q.2 $40$ $2$ $2$ $1$ $0$ 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
40.480.33.cx.2 $40$ $5$ $5$ $33$ $3$ $1^{14}\cdot2^{9}$
40.576.33.jz.2 $40$ $6$ $6$ $33$ $3$ $1^{14}\cdot2\cdot4^{4}$
40.960.65.nt.1 $40$ $10$ $10$ $65$ $5$ $1^{28}\cdot2^{10}\cdot4^{4}$
80.192.5.g.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.m.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.bi.2 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.bk.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.ep.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.er.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.fn.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.5.ft.1 $80$ $2$ $2$ $5$ $?$ not computed
80.192.9.jp.2 $80$ $2$ $2$ $9$ $?$ not computed
80.192.9.jq.2 $80$ $2$ $2$ $9$ $?$ not computed
80.192.9.jv.2 $80$ $2$ $2$ $9$ $?$ not computed
80.192.9.jw.2 $80$ $2$ $2$ $9$ $?$ not computed
120.288.17.cae.2 $120$ $3$ $3$ $17$ $?$ not computed
120.384.17.to.2 $120$ $4$ $4$ $17$ $?$ not computed
240.192.5.be.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.bk.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.du.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.dw.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.of.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.oh.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.qr.1 $240$ $2$ $2$ $5$ $?$ not computed
240.192.5.qx.2 $240$ $2$ $2$ $5$ $?$ not computed
240.192.9.bjj.2 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.bjk.2 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.bjt.2 $240$ $2$ $2$ $9$ $?$ not computed
240.192.9.bju.2 $240$ $2$ $2$ $9$ $?$ not computed