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Properties

Label 40.96.1.bw.1
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 $2^{2}\cdot4^{3}$
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: 40.96.1.540

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}7&8\\6&1\end{bmatrix}$, $\begin{bmatrix}13&4\\16&25\end{bmatrix}$, $\begin{bmatrix}23&36\\18&27\end{bmatrix}$, $\begin{bmatrix}29&24\\30&13\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.192.1-40.bw.1.1, 40.192.1-40.bw.1.2, 40.192.1-40.bw.1.3, 40.192.1-40.bw.1.4, 40.192.1-40.bw.1.5, 40.192.1-40.bw.1.6, 40.192.1-40.bw.1.7, 40.192.1-40.bw.1.8, 120.192.1-40.bw.1.1, 120.192.1-40.bw.1.2, 120.192.1-40.bw.1.3, 120.192.1-40.bw.1.4, 120.192.1-40.bw.1.5, 120.192.1-40.bw.1.6, 120.192.1-40.bw.1.7, 120.192.1-40.bw.1.8, 280.192.1-40.bw.1.1, 280.192.1-40.bw.1.2, 280.192.1-40.bw.1.3, 280.192.1-40.bw.1.4, 280.192.1-40.bw.1.5, 280.192.1-40.bw.1.6, 280.192.1-40.bw.1.7, 280.192.1-40.bw.1.8
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} $
$=$ $2 x^{2} + y^{2} + 2 y w + 4 z^{2} - 2 w^{2}$
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Singular plane model Singular plane model

$ 0 $ $=$ $ 9 x^{4} + 6 x^{2} y^{2} - 8 x^{2} z^{2} + y^{4} - 6 y^{2} z^{2} + 4 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:

$\displaystyle X$ $=$ $\displaystyle z$
$\displaystyle Y$ $=$ $\displaystyle 2x$
$\displaystyle Z$ $=$ $\displaystyle w$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle 2^4\,\frac{1875384yz^{22}w-67469976yz^{20}w^{3}+826796880yz^{18}w^{5}-4993124976yz^{16}w^{7}+17266233600yz^{14}w^{9}-36933361920yz^{12}w^{11}+50926256640yz^{10}w^{13}-46019538432yz^{8}w^{15}+27064074240yz^{6}w^{17}-9970974720yz^{4}w^{19}+2089156608yz^{2}w^{21}-189923328yw^{23}+226981z^{24}-20369364z^{22}w^{2}+385503084z^{20}w^{4}-3157793720z^{18}w^{6}+14005732128z^{16}w^{8}-37436023680z^{14}w^{10}+63964584320z^{12}w^{12}-72096188160z^{10}w^{14}+54161410560z^{8}w^{16}-26812702720z^{6}w^{18}+8391979008z^{4}w^{20}-1503510528z^{2}w^{22}+117379072w^{24}}{z^{8}(15120yz^{14}w-328272yz^{12}w^{3}+1985760yz^{10}w^{5}-5210016yz^{8}w^{7}+6948480yz^{6}w^{9}-4927104yz^{4}w^{11}+1768704yz^{2}w^{13}-252672yw^{15}+2025z^{16}-130680z^{14}w^{2}+1311448z^{12}w^{4}-4933520z^{10}w^{6}+9046448z^{8}w^{8}-8955584z^{6}w^{10}+4881344z^{4}w^{12}-1375616z^{2}w^{14}+156160w^{16})}$

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.h.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.j.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.l.2 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.0.ba.1 $40$ $2$ $2$ $0$ $0$ full Jacobian
40.48.1.bh.2 $40$ $2$ $2$ $1$ $0$ dimension zero
40.48.1.bj.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.la.2 $40$ $5$ $5$ $33$ $7$ $1^{14}\cdot2^{9}$
40.576.33.wc.2 $40$ $6$ $6$ $33$ $5$ $1^{14}\cdot2\cdot4^{4}$
40.960.65.bgq.1 $40$ $10$ $10$ $65$ $9$ $1^{28}\cdot2^{10}\cdot4^{4}$
120.288.17.juu.2 $120$ $3$ $3$ $17$ $?$ not computed
120.384.17.drs.2 $120$ $4$ $4$ $17$ $?$ not computed