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Properties

Label 40.192.1-40.a.1.12
Level $40$
Index $192$
Genus $1$
Analytic rank $0$
Cusps $16$
$\Q$-cusps $0$

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Invariants

Level: $40$ $\SL_2$-level: $8$ Newform level: $1600$
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 $2^{2}\cdot4^{3}$
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: 40.192.1.269

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}1&8\\10&3\end{bmatrix}$, $\begin{bmatrix}21&28\\10&31\end{bmatrix}$, $\begin{bmatrix}27&20\\20&11\end{bmatrix}$, $\begin{bmatrix}33&12\\24&13\end{bmatrix}$
Contains $-I$: no $\quad$ (see 40.96.1.a.1 for the level structure with $-I$)
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $192$
Full 40-torsion field degree: $3840$

Jacobian

Conductor: $2^{6}\cdot5^{2}$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 1600.2.a.n

Models

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

$ 0 $ $=$ $ x^{2} + 2 y^{2} + z^{2} $
$=$ $3 x^{2} + x z - 2 x w - 2 z^{2} - 2 z w + 2 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 13 x^{4} + 8 x^{3} z + 31 x^{2} y^{2} - 6 x^{2} z^{2} + 8 x y^{2} z - 4 x z^{3} + 18 y^{4} - 8 y^{2} z^{2} + 2 z^{4} $
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Rational points

This modular curve has no real points, and therefore no 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{1}{3^4}\cdot\frac{228255878482099xz^{23}-119199109414702xz^{22}w-1063354766461328xz^{21}w^{2}+539001154514128xz^{20}w^{3}+1228815074753720xz^{19}w^{4}-1694320898086256xz^{18}w^{5}-354248696980032xz^{17}w^{6}+1852987320115584xz^{16}w^{7}-482585374318848xz^{15}w^{8}-794729866831360xz^{14}w^{9}+451435281919744xz^{13}w^{10}+81415674278912xz^{12}w^{11}-150730458701056xz^{11}w^{12}+42762903591424xz^{10}w^{13}+19876847836160xz^{9}w^{14}-15192956608512xz^{8}w^{15}-809776844544xz^{7}w^{16}+3272432936448xz^{6}w^{17}-927240507392xz^{5}w^{18}-48055029760xz^{4}w^{19}+97096050688xz^{3}w^{20}-34674987008xz^{2}w^{21}+6934839296xzw^{22}-603029504xw^{23}-157354581993293z^{24}-119199109414702z^{23}w+696273092427714z^{22}w^{2}-117170345275608z^{21}w^{3}-1918203216260812z^{20}w^{4}+725336315570352z^{19}w^{5}+2203943518803728z^{18}w^{6}-1317906107817024z^{17}w^{7}-948873757134960z^{16}w^{8}+1100156286597632z^{15}w^{9}-109025564384768z^{14}w^{10}-420486513427968z^{13}w^{11}+230653947197952z^{12}w^{12}+64867038141952z^{11}w^{13}-92512416907776z^{10}w^{14}+16263651198976z^{9}w^{15}+10255988668416z^{8}w^{16}-4866194631168z^{7}w^{17}+123222398464z^{6}w^{18}+525975861248z^{5}w^{19}-132688892928z^{4}w^{20}-55104073728z^{3}w^{21}+40458235904z^{2}w^{22}-9247506432zw^{23}+770625536w^{24}}{z^{4}(3570125xz^{19}-1208350xz^{18}w-33081750xz^{17}w^{2}+17421924xz^{16}w^{3}+125255775xz^{15}w^{4}-81516510xz^{14}w^{5}-255244110xz^{13}w^{6}+181791768xz^{12}w^{7}+39865365xz^{11}w^{8}+332030350xz^{10}w^{9}-238125800xz^{9}w^{10}-731967408xz^{8}w^{11}+928078320xz^{7}w^{12}-47274720xz^{6}w^{13}-485956800xz^{5}w^{14}+282026496xz^{4}w^{15}-11942160xz^{3}w^{16}-39489120xz^{2}w^{17}+13631360xzw^{18}-1434880xw^{19}+3570125z^{20}-1208350z^{19}w-34509800z^{18}w^{2}+18476484z^{17}w^{3}+137723919z^{16}w^{4}-93082350z^{15}w^{5}-297966060z^{14}w^{6}+227736288z^{13}w^{7}+591485181z^{12}w^{8}-519415610z^{11}w^{9}-892244690z^{10}w^{10}+1202249192z^{9}w^{11}+217176468z^{8}w^{12}-1038168480z^{7}w^{13}+492931680z^{6}w^{14}+115664256z^{5}w^{15}-185869440z^{4}w^{16}+71175840z^{3}w^{17}-13840480z^{2}w^{18}+1873280zw^{19}-187328w^{20})}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 40.96.1.a.1 :

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

Equation of the image curve:

$0$ $=$ $ 13X^{4}+31X^{2}Y^{2}+18Y^{4}+8X^{3}Z+8XY^{2}Z-6X^{2}Z^{2}-8Y^{2}Z^{2}-4XZ^{3}+2Z^{4} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
8.96.0-8.a.1.7 $8$ $2$ $2$ $0$ $0$ full Jacobian
40.96.0-8.a.1.6 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.96.0-40.b.2.7 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.96.0-40.b.2.20 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.96.1-40.n.2.18 $40$ $2$ $2$ $1$ $0$ dimension zero
40.96.1-40.n.2.23 $40$ $2$ $2$ $1$ $0$ 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
40.384.5-40.a.1.4 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.384.5-40.a.1.6 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.384.5-40.c.2.2 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.384.5-40.c.2.5 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.384.5-40.d.1.3 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.384.5-40.d.1.5 $40$ $2$ $2$ $5$ $0$ $1^{2}\cdot2$
40.384.5-40.f.4.4 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.384.5-40.f.4.6 $40$ $2$ $2$ $5$ $1$ $1^{2}\cdot2$
40.960.33-40.bt.2.9 $40$ $5$ $5$ $33$ $9$ $1^{14}\cdot2^{9}$
40.1152.33-40.gv.1.19 $40$ $6$ $6$ $33$ $4$ $1^{14}\cdot2\cdot4^{4}$
40.1920.65-40.ir.1.19 $40$ $10$ $10$ $65$ $12$ $1^{28}\cdot2^{10}\cdot4^{4}$
120.384.5-120.l.1.8 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.l.1.13 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.n.2.8 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.n.2.9 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.u.2.6 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.u.2.15 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.x.2.6 $120$ $2$ $2$ $5$ $?$ not computed
120.384.5-120.x.2.11 $120$ $2$ $2$ $5$ $?$ not computed
280.384.5-280.h.2.6 $280$ $2$ $2$ $5$ $?$ not computed
280.384.5-280.h.2.15 $280$ $2$ $2$ $5$ $?$ not computed
280.384.5-280.i.2.6 $280$ $2$ $2$ $5$ $?$ not computed
280.384.5-280.i.2.11 $280$ $2$ $2$ $5$ $?$ not computed
280.384.5-280.m.1.8 $280$ $2$ $2$ $5$ $?$ not computed
280.384.5-280.m.1.13 $280$ $2$ $2$ $5$ $?$ not computed
280.384.5-280.o.2.8 $280$ $2$ $2$ $5$ $?$ not computed
280.384.5-280.o.2.9 $280$ $2$ $2$ $5$ $?$ not computed