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Properties

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

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Invariants

Level: $40$ $\SL_2$-level: $8$ Newform level: $800$
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: $1$
$\Q$-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: 40.96.1.174

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}7&36\\32&3\end{bmatrix}$, $\begin{bmatrix}17&4\\12&5\end{bmatrix}$, $\begin{bmatrix}25&4\\24&7\end{bmatrix}$, $\begin{bmatrix}27&28\\20&17\end{bmatrix}$, $\begin{bmatrix}33&20\\12&31\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.192.1-40.o.1.1, 40.192.1-40.o.1.2, 40.192.1-40.o.1.3, 40.192.1-40.o.1.4, 40.192.1-40.o.1.5, 40.192.1-40.o.1.6, 40.192.1-40.o.1.7, 40.192.1-40.o.1.8, 40.192.1-40.o.1.9, 40.192.1-40.o.1.10, 40.192.1-40.o.1.11, 40.192.1-40.o.1.12, 40.192.1-40.o.1.13, 40.192.1-40.o.1.14, 40.192.1-40.o.1.15, 40.192.1-40.o.1.16, 120.192.1-40.o.1.1, 120.192.1-40.o.1.2, 120.192.1-40.o.1.3, 120.192.1-40.o.1.4, 120.192.1-40.o.1.5, 120.192.1-40.o.1.6, 120.192.1-40.o.1.7, 120.192.1-40.o.1.8, 120.192.1-40.o.1.9, 120.192.1-40.o.1.10, 120.192.1-40.o.1.11, 120.192.1-40.o.1.12, 120.192.1-40.o.1.13, 120.192.1-40.o.1.14, 120.192.1-40.o.1.15, 120.192.1-40.o.1.16, 280.192.1-40.o.1.1, 280.192.1-40.o.1.2, 280.192.1-40.o.1.3, 280.192.1-40.o.1.4, 280.192.1-40.o.1.5, 280.192.1-40.o.1.6, 280.192.1-40.o.1.7, 280.192.1-40.o.1.8, 280.192.1-40.o.1.9, 280.192.1-40.o.1.10, 280.192.1-40.o.1.11, 280.192.1-40.o.1.12, 280.192.1-40.o.1.13, 280.192.1-40.o.1.14, 280.192.1-40.o.1.15, 280.192.1-40.o.1.16
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $96$
Full 40-torsion field degree: $7680$

Jacobian

Conductor: $2^{5}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 800.2.a.d

Models

Embedded model Embedded model in $\mathbb{P}^{3}$

$ 0 $ $=$ $ 2 x^{2} - y^{2} - z^{2} $
$=$ $20 x y - 10 y z + w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 200 x^{4} + x^{2} y^{2} + 2 x y 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:

$\displaystyle X$ $=$ $\displaystyle y$
$\displaystyle Y$ $=$ $\displaystyle 10z$
$\displaystyle Z$ $=$ $\displaystyle w$

Maps to other modular curves

$j$-invariant map of degree 96 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle 2^2\,\frac{69547484160000000000000xz^{23}+709223864332500000000xz^{19}w^{4}+2767397267000000000xz^{15}w^{8}+5202774492000000xz^{11}w^{12}+4751376680000xz^{7}w^{16}+1747407600xz^{3}w^{20}-28807511167968750000000y^{2}z^{22}-279293639786250000000y^{2}z^{18}w^{4}-1026700400850000000y^{2}z^{14}w^{8}-1790741458000000y^{2}z^{10}w^{12}-1470817385000y^{2}z^{6}w^{16}-440146200y^{2}z^{2}w^{20}-1193246182331250000000yz^{21}w^{2}-10969084976750000000yz^{17}w^{6}-37874945550000000yz^{13}w^{10}-60990463600000yz^{9}w^{14}-44610807000yz^{5}w^{18}-9592200yzw^{22}-49177497664000000000000z^{24}-543684627194062500000z^{20}w^{4}-2326565468056250000z^{16}w^{8}-4885362274500000z^{12}w^{12}-5163958187500z^{8}w^{16}-2414167350z^{4}w^{20}-389017w^{24}}{w^{8}(1506662400000000xz^{15}+9307395000000xz^{11}w^{4}+14561080000xz^{7}w^{8}+3986400xz^{3}w^{12}-624079987500000y^{2}z^{14}-3541645500000y^{2}z^{10}w^{4}-4701670000y^{2}z^{6}w^{8}-766800y^{2}z^{2}w^{12}-25850222500000yz^{13}w^{2}-133711300000yz^{9}w^{6}-146258000yz^{5}w^{10}-10800yzw^{14}-1065371200000000z^{16}-7495262625000z^{12}w^{4}-14631692500z^{8}w^{8}-6695900z^{4}w^{12}-81w^{16})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.48.0.b.2 $8$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.b.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.1.o.1 $40$ $2$ $2$ $1$ $1$ 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
40.192.5.e.1 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.192.5.f.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.192.5.i.1 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.192.5.k.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.192.5.q.1 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.192.5.s.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.192.5.ba.1 $40$ $2$ $2$ $5$ $2$ $1^{2}\cdot2$
40.192.5.bb.1 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.480.33.ck.2 $40$ $5$ $5$ $33$ $6$ $1^{14}\cdot2^{9}$
40.576.33.ir.1 $40$ $6$ $6$ $33$ $4$ $1^{14}\cdot2\cdot4^{4}$
40.960.65.lq.2 $40$ $10$ $10$ $65$ $8$ $1^{28}\cdot2^{10}\cdot4^{4}$
120.192.5.bv.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.bx.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.dp.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.dr.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ev.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.ex.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.hb.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.hd.1 $120$ $2$ $2$ $5$ $?$ not computed
120.288.17.btf.2 $120$ $3$ $3$ $17$ $?$ not computed
120.384.17.pt.2 $120$ $4$ $4$ $17$ $?$ not computed
280.192.5.cm.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.cn.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.dp.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.dq.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.ev.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.ew.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.fw.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.fx.1 $280$ $2$ $2$ $5$ $?$ not computed