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Properties

Label 40.96.1.bs.2
Level $40$
Index $96$
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: $96$ $\PSL_2$-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 $4^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2 \le \gamma \le 4$
$\overline{\Q}$-gonality: $2$
Rational cusps: $0$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8K1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.96.1.476

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}7&0\\28&23\end{bmatrix}$, $\begin{bmatrix}7&16\\38&23\end{bmatrix}$, $\begin{bmatrix}11&32\\38&29\end{bmatrix}$, $\begin{bmatrix}35&12\\12&7\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.192.1-40.bs.2.1, 40.192.1-40.bs.2.2, 40.192.1-40.bs.2.3, 40.192.1-40.bs.2.4, 40.192.1-40.bs.2.5, 40.192.1-40.bs.2.6, 40.192.1-40.bs.2.7, 40.192.1-40.bs.2.8, 120.192.1-40.bs.2.1, 120.192.1-40.bs.2.2, 120.192.1-40.bs.2.3, 120.192.1-40.bs.2.4, 120.192.1-40.bs.2.5, 120.192.1-40.bs.2.6, 120.192.1-40.bs.2.7, 120.192.1-40.bs.2.8, 280.192.1-40.bs.2.1, 280.192.1-40.bs.2.2, 280.192.1-40.bs.2.3, 280.192.1-40.bs.2.4, 280.192.1-40.bs.2.5, 280.192.1-40.bs.2.6, 280.192.1-40.bs.2.7, 280.192.1-40.bs.2.8
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $192$
Full 40-torsion field degree: $7680$

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

$ 0 $ $=$ $ x^{4} - 6 x^{2} y^{2} + 3 x^{2} z^{2} + 9 y^{4} - 4 y^{2} z^{2} + z^{4} $
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Rational points

This modular curve has no real points, and therefore no rational points.

Maps between models of this curve

Birational map from embedded model to plane model:

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

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle -\frac{2^4}{3^8}\cdot\frac{15819935688yz^{22}w-2053015632yz^{20}w^{3}-185379217920yz^{18}w^{5}-180688050432yz^{16}w^{7}+403618723200yz^{14}w^{9}+1083756360960yz^{12}w^{11}+1167579463680yz^{10}w^{13}+733976764416yz^{8}w^{15}+290019502080yz^{6}w^{17}+71232860160yz^{4}w^{19}+9949986816yz^{2}w^{21}+603029504yw^{23}+40209003207z^{24}-217027041228z^{22}w^{2}-5029478892z^{20}w^{4}+855747555840z^{18}w^{6}+734670966816z^{16}w^{8}-929914032960z^{14}w^{10}-2361533952960z^{12}w^{12}-2270183915520z^{10}w^{14}-1283814074880z^{8}w^{16}-461939235840z^{6}w^{18}-104487312384z^{4}w^{20}-13569744896z^{2}w^{22}-770625536w^{24}}{z^{8}(43740yz^{14}w+376164yz^{12}w^{3}+1268460yz^{10}w^{5}+2124792yz^{8}w^{7}+1877040yz^{6}w^{9}+887328yz^{4}w^{11}+212352yz^{2}w^{13}+20224yw^{15}-54675z^{16}-575910z^{14}w^{2}-2499741z^{12}w^{4}-5717790z^{10}w^{6}-7345971z^{8}w^{8}-5315112z^{6}w^{10}-2143128z^{4}w^{12}-449728z^{2}w^{14}-38320w^{16})}$

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
8.48.0.g.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.i.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.j.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.z.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.1.bg.1 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.bh.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.bs.1 $40$ $2$ $2$ $1$ $0$ 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.kw.2 $40$ $5$ $5$ $33$ $9$ $1^{14}\cdot2^{9}$
40.576.33.vu.1 $40$ $6$ $6$ $33$ $5$ $1^{14}\cdot2\cdot4^{4}$
40.960.65.bge.2 $40$ $10$ $10$ $65$ $11$ $1^{28}\cdot2^{10}\cdot4^{4}$
120.288.17.jtu.2 $120$ $3$ $3$ $17$ $?$ not computed
120.384.17.dri.1 $120$ $4$ $4$ $17$ $?$ not computed