↑

Properties

Label 40.96.3.be.1
Level $40$
Index $96$
Genus $3$
Analytic rank $0$
Cusps $12$
$\Q$-cusps $4$

Related objects

Downloads

Learn more

Invariants

Level: $40$ $\SL_2$-level: $8$ Newform level: $1600$
Index: $96$ $\PSL_2$-index:$96$
Genus: $3 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (of which $4$ are rational) Cusp widths $8^{12}$ Cusp orbits $1^{4}\cdot2^{2}\cdot4$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $4$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 8B3
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 40.96.3.5

Level structure

$\GL_2(\Z/40\Z)$-generators: $\begin{bmatrix}1&34\\16&23\end{bmatrix}$, $\begin{bmatrix}3&22\\4&29\end{bmatrix}$, $\begin{bmatrix}11&10\\4&13\end{bmatrix}$, $\begin{bmatrix}33&14\\0&27\end{bmatrix}$, $\begin{bmatrix}37&2\\4&23\end{bmatrix}$
Contains $-I$: yes
Quadratic refinements: 40.192.3-40.be.1.1, 40.192.3-40.be.1.2, 40.192.3-40.be.1.3, 40.192.3-40.be.1.4, 40.192.3-40.be.1.5, 40.192.3-40.be.1.6, 40.192.3-40.be.1.7, 40.192.3-40.be.1.8, 40.192.3-40.be.1.9, 40.192.3-40.be.1.10, 40.192.3-40.be.1.11, 40.192.3-40.be.1.12, 80.192.3-40.be.1.1, 80.192.3-40.be.1.2, 80.192.3-40.be.1.3, 80.192.3-40.be.1.4, 80.192.3-40.be.1.5, 80.192.3-40.be.1.6, 80.192.3-40.be.1.7, 80.192.3-40.be.1.8, 120.192.3-40.be.1.1, 120.192.3-40.be.1.2, 120.192.3-40.be.1.3, 120.192.3-40.be.1.4, 120.192.3-40.be.1.5, 120.192.3-40.be.1.6, 120.192.3-40.be.1.7, 120.192.3-40.be.1.8, 120.192.3-40.be.1.9, 120.192.3-40.be.1.10, 120.192.3-40.be.1.11, 120.192.3-40.be.1.12, 240.192.3-40.be.1.1, 240.192.3-40.be.1.2, 240.192.3-40.be.1.3, 240.192.3-40.be.1.4, 240.192.3-40.be.1.5, 240.192.3-40.be.1.6, 240.192.3-40.be.1.7, 240.192.3-40.be.1.8, 280.192.3-40.be.1.1, 280.192.3-40.be.1.2, 280.192.3-40.be.1.3, 280.192.3-40.be.1.4, 280.192.3-40.be.1.5, 280.192.3-40.be.1.6, 280.192.3-40.be.1.7, 280.192.3-40.be.1.8, 280.192.3-40.be.1.9, 280.192.3-40.be.1.10, 280.192.3-40.be.1.11, 280.192.3-40.be.1.12
Cyclic 40-isogeny field degree: $12$
Cyclic 40-torsion field degree: $96$
Full 40-torsion field degree: $7680$

Jacobian

Conductor: $2^{18}\cdot5^{6}$
Simple: no
Squarefree: yes
Decomposition: $1\cdot2$
Newforms: 1600.2.a.n, 1600.2.d.a

Models

Embedded model Embedded model in $\mathbb{P}^{4}$

$ 0 $ $=$ $ 2 x z t - x w t - y w t - y t^{2} $
$=$ $2 x z w - x w^{2} - y w^{2} - y w t$
$=$ $2 x z^{2} - x z w - y z w - y z t$
$=$ $2 x y z - x y w - y^{2} w - y^{2} t$
$=$$\cdots$
Copy content Toggle raw display

Singular plane model Singular plane model

$ 0 $ $=$ $ - x^{4} z + 5 x^{3} y^{2} - 5 x^{3} z^{2} + 10 x^{2} y^{2} z - 6 x^{2} z^{3} - 10 x y^{2} z^{2} + \cdots - 20 y^{2} z^{3} $
 Copy content Toggle raw display

Weierstrass model Weierstrass model

$ y^{2} $ $=$ $ -5x^{7} + 35x^{5} - 35x^{3} + 5x $
 Copy content Toggle raw display

Rational points

This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Embedded model
$(0:0:-1/2:-1:1)$, $(0:1:0:0:0)$, $(1:0:0:0:0)$, $(-1:1:0:0:0)$

Maps between models of this curve

Birational map from embedded model to plane model:

$\displaystyle X$ $=$ $\displaystyle x$
$\displaystyle Y$ $=$ $\displaystyle \frac{1}{5}t$
$\displaystyle Z$ $=$ $\displaystyle y$

Birational map from embedded model to Weierstrass model:

$\displaystyle X$ $=$ $\displaystyle -x-y$
$\displaystyle Y$ $=$ $\displaystyle x^{3}t+2x^{2}yt-2xy^{2}t-4y^{3}t$
$\displaystyle Z$ $=$ $\displaystyle y$

Maps to other modular curves

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

$\displaystyle j$ $=$ $\displaystyle \frac{78125x^{14}+93750x^{12}t^{2}+68750x^{10}t^{4}-15000x^{8}t^{6}+142750x^{6}t^{8}-280800x^{4}t^{10}+624540x^{2}t^{12}+156250xy^{13}+718750xy^{11}t^{2}+175000xy^{9}t^{4}+775000xy^{7}t^{6}+849250xy^{5}t^{8}+2752550xy^{3}t^{10}+6047680xyt^{12}+312500y^{12}t^{2}-50000y^{10}t^{4}+555000y^{8}t^{6}+1048000y^{6}t^{8}+2574700y^{4}t^{10}+6906400y^{2}t^{12}-640zw^{13}+128zw^{12}t-4352zw^{11}t^{2}+256zw^{10}t^{3}-43904zw^{9}t^{4}+4480zw^{8}t^{5}-232960zw^{7}t^{6}+382152zw^{6}t^{7}-1223424zw^{5}t^{8}+2461408zw^{4}t^{9}-3658176zw^{3}t^{10}+5776040zw^{2}t^{11}-4838272zwt^{12}+640zt^{13}+224w^{14}-64w^{13}t+1568w^{12}t^{2}-384w^{11}t^{3}+15456w^{10}t^{4}-4032w^{9}t^{5}+83104w^{8}t^{6}-153428w^{7}t^{7}+430884w^{6}t^{8}-1026920w^{5}t^{9}+1319744w^{4}t^{10}-2783260w^{3}t^{11}+1862948w^{2}t^{12}-1631160wt^{13}+224t^{14}}{t^{4}(125x^{6}t^{4}-250x^{4}t^{6}+590x^{2}t^{8}+250xy^{5}t^{4}-50xy^{3}t^{6}+2960xyt^{8}+500y^{4}t^{6}+1360y^{2}t^{8}-40zw^{9}+8zw^{8}t-272zw^{7}t^{2}+16zw^{6}t^{3}-904zw^{5}t^{4}-88zw^{4}t^{5}-2048zw^{3}t^{6}-184zw^{2}t^{7}-2368zwt^{8}+14w^{10}-4w^{9}t+98w^{8}t^{2}-24w^{7}t^{3}+322w^{6}t^{4}-68w^{5}t^{5}+686w^{4}t^{6}-236w^{3}t^{7}+732w^{2}t^{8}-508wt^{9})}$

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
8.48.0.c.1 $8$ $2$ $2$ $0$ $0$ full Jacobian
40.48.1.n.1 $40$ $2$ $2$ $1$ $0$ $2$
40.48.2.a.1 $40$ $2$ $2$ $2$ $0$ $1$

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.z.1 $40$ $2$ $2$ $5$ $0$ $1^{2}$
40.192.5.z.2 $40$ $2$ $2$ $5$ $0$ $1^{2}$
40.192.5.bb.3 $40$ $2$ $2$ $5$ $1$ $1^{2}$
40.192.5.bb.4 $40$ $2$ $2$ $5$ $1$ $1^{2}$
40.480.35.bo.1 $40$ $5$ $5$ $35$ $5$ $1^{14}\cdot2^{5}\cdot4^{2}$
40.576.37.es.1 $40$ $6$ $6$ $37$ $4$ $1^{14}\cdot2^{2}\cdot4^{4}$
40.960.69.gu.1 $40$ $10$ $10$ $69$ $8$ $1^{28}\cdot2^{7}\cdot4^{6}$
80.192.7.d.1 $80$ $2$ $2$ $7$ $?$ not computed
80.192.7.g.1 $80$ $2$ $2$ $7$ $?$ not computed
80.192.7.z.1 $80$ $2$ $2$ $7$ $?$ not computed
80.192.7.ba.1 $80$ $2$ $2$ $7$ $?$ not computed
80.192.7.cj.1 $80$ $2$ $2$ $7$ $?$ not computed
80.192.7.ck.1 $80$ $2$ $2$ $7$ $?$ not computed
80.192.7.dd.1 $80$ $2$ $2$ $7$ $?$ not computed
80.192.7.dg.1 $80$ $2$ $2$ $7$ $?$ not computed
120.192.5.hl.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.hl.2 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.hn.1 $120$ $2$ $2$ $5$ $?$ not computed
120.192.5.hn.2 $120$ $2$ $2$ $5$ $?$ not computed
120.288.19.pw.1 $120$ $3$ $3$ $19$ $?$ not computed
120.384.21.iw.1 $120$ $4$ $4$ $21$ $?$ not computed
240.192.7.u.1 $240$ $2$ $2$ $7$ $?$ not computed
240.192.7.x.1 $240$ $2$ $2$ $7$ $?$ not computed
240.192.7.ck.1 $240$ $2$ $2$ $7$ $?$ not computed
240.192.7.cl.1 $240$ $2$ $2$ $7$ $?$ not computed
240.192.7.fq.1 $240$ $2$ $2$ $7$ $?$ not computed
240.192.7.fr.1 $240$ $2$ $2$ $7$ $?$ not computed
240.192.7.he.1 $240$ $2$ $2$ $7$ $?$ not computed
240.192.7.hh.1 $240$ $2$ $2$ $7$ $?$ not computed
280.192.5.gz.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.gz.2 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.ha.1 $280$ $2$ $2$ $5$ $?$ not computed
280.192.5.ha.2 $280$ $2$ $2$ $5$ $?$ not computed