↑

Properties

Label 280.96.2-28.b.1.15
Level $280$
Index $96$
Genus $2$
Cusps $6$
$\Q$-cusps $2$

Related objects

Downloads

Learn more

Invariants

Level: $280$ $\SL_2$-level: $56$ Newform level: $112$
Index: $96$ $\PSL_2$-index:$48$
Genus: $2 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$
Cusps: $6$ (of which $2$ are rational) Cusp widths $1^{2}\cdot4\cdot7^{2}\cdot28$ Cusp orbits $1^{2}\cdot2^{2}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: not computed
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $2$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 28D2

Level structure

$\GL_2(\Z/280\Z)$-generators: $\begin{bmatrix}46&159\\259&268\end{bmatrix}$, $\begin{bmatrix}56&13\\97&42\end{bmatrix}$, $\begin{bmatrix}110&161\\183&144\end{bmatrix}$, $\begin{bmatrix}146&81\\219&260\end{bmatrix}$, $\begin{bmatrix}256&169\\149&94\end{bmatrix}$, $\begin{bmatrix}274&131\\159&64\end{bmatrix}$
Contains $-I$: no $\quad$ (see 28.48.2.b.1 for the level structure with $-I$)
Cyclic 280-isogeny field degree: $24$
Cyclic 280-torsion field degree: $2304$
Full 280-torsion field degree: $15482880$

Models

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

$ 0 $ $=$ $ x z^{2} + 2 y^{2} z + y z w $
$=$ $x y z + 2 y^{3} + y^{2} w$
$=$ $x z w + 2 y^{2} w + y w^{2}$
$=$ $x^{2} y + x y z - y^{3} - 2 y^{2} w + y z^{2} + y w^{2} + z^{2} w$
$=$$\cdots$
Copy content Toggle raw display

Singular plane model Singular plane model

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

Weierstrass model Weierstrass model

$ y^{2} + \left(x^{3} + x\right) y $ $=$ $ -3x^{4} + 6x^{2} - 7 $
 Copy content Toggle raw display

Rational points

This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Embedded model
$(0:0:1:0)$, $(0:0:0:1)$

Maps to other modular curves

$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{1560062x^{2}z^{8}-1419110x^{2}z^{6}w^{2}-5884198x^{2}z^{4}w^{4}-1660602x^{2}z^{2}w^{6}+153664x^{2}w^{8}-1529276xyz^{7}w+6017344xyz^{5}w^{3}+3177332xyz^{3}w^{5}-1684536xyzw^{7}+1182468xz^{9}+1667533xz^{7}w^{2}+54521xz^{5}w^{4}-2911133xz^{3}w^{6}-1140769xzw^{8}-5545932yz^{8}w-2854219yz^{6}w^{3}+12710201yz^{4}w^{5}+6411883yz^{2}w^{7}-372449yw^{9}+1229312z^{10}-675528z^{8}w^{2}+2010368z^{6}w^{4}+5328776z^{4}w^{6}+615168z^{2}w^{8}}{98x^{2}z^{8}-1442x^{2}z^{6}w^{2}+2102x^{2}z^{4}w^{4}-14x^{2}z^{2}w^{6}-476xyz^{7}w-5400xyz^{5}w^{3}+1668xyz^{3}w^{5}-112xyzw^{7}-196xz^{9}-4249xz^{7}w^{2}-3985xz^{5}w^{4}+833xz^{3}w^{6}+49xzw^{8}+2044yz^{8}w+5967yz^{6}w^{3}-3393yz^{4}w^{5}+1001yz^{2}w^{7}+49yw^{9}+1848z^{8}w^{2}+640z^{6}w^{4}+168z^{4}w^{6}}$

Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 28.48.2.b.1 :

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

Equation of the image curve:

$0$ $=$ $ 4X^{5}+4X^{4}Y+X^{3}Y^{2}-3X^{3}Z^{2}-3X^{2}YZ^{2}+XY^{2}Z^{2}+XZ^{4}+YZ^{4} $

Map of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve 28.48.2.b.1 :

$\displaystyle X$ $=$ $\displaystyle z^{2}$
$\displaystyle Y$ $=$ $\displaystyle -2y^{4}z^{2}-y^{3}z^{2}w+y^{2}z^{4}-yz^{4}w-z^{6}$
$\displaystyle Z$ $=$ $\displaystyle -yz$

Modular covers

The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank
40.12.0-4.b.1.3 $40$ $8$ $8$ $0$ $0$
$X_0(7)$ $7$ $12$ $6$ $0$ $0$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
40.12.0-4.b.1.3 $40$ $8$ $8$ $0$ $0$

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus
280.192.5-28.a.1.8 $280$ $2$ $2$ $5$
280.192.5-28.e.1.5 $280$ $2$ $2$ $5$
280.192.5-28.i.1.5 $280$ $2$ $2$ $5$
280.192.5-28.j.1.5 $280$ $2$ $2$ $5$
280.288.4-28.c.1.13 $280$ $3$ $3$ $4$
280.288.4-28.c.2.15 $280$ $3$ $3$ $4$
280.288.4-28.e.1.15 $280$ $3$ $3$ $4$
280.192.4-56.c.1.3 $280$ $2$ $2$ $4$
280.192.4-56.c.1.4 $280$ $2$ $2$ $4$
280.192.4-56.c.2.3 $280$ $2$ $2$ $4$
280.192.4-56.c.2.4 $280$ $2$ $2$ $4$
280.192.4-56.d.1.3 $280$ $2$ $2$ $4$
280.192.4-56.d.1.4 $280$ $2$ $2$ $4$
280.192.4-56.d.2.3 $280$ $2$ $2$ $4$
280.192.4-56.d.2.4 $280$ $2$ $2$ $4$
280.192.5-56.e.1.12 $280$ $2$ $2$ $5$
280.192.5-56.n.1.11 $280$ $2$ $2$ $5$
280.192.5-56.y.1.11 $280$ $2$ $2$ $5$
280.192.5-56.bb.1.11 $280$ $2$ $2$ $5$
280.192.6-56.c.1.3 $280$ $2$ $2$ $6$
280.192.6-56.c.1.4 $280$ $2$ $2$ $6$
280.192.6-56.c.2.3 $280$ $2$ $2$ $6$
280.192.6-56.c.2.4 $280$ $2$ $2$ $6$
280.192.6-56.d.1.2 $280$ $2$ $2$ $6$
280.192.6-56.d.1.4 $280$ $2$ $2$ $6$
280.192.6-56.d.2.2 $280$ $2$ $2$ $6$
280.192.6-56.d.2.4 $280$ $2$ $2$ $6$
280.192.5-140.i.1.12 $280$ $2$ $2$ $5$
280.192.5-140.j.1.10 $280$ $2$ $2$ $5$
280.192.5-140.m.1.16 $280$ $2$ $2$ $5$
280.192.5-140.n.1.14 $280$ $2$ $2$ $5$
280.480.18-140.b.1.1 $280$ $5$ $5$ $18$
280.192.4-280.c.1.19 $280$ $2$ $2$ $4$
280.192.4-280.c.1.20 $280$ $2$ $2$ $4$
280.192.4-280.c.2.21 $280$ $2$ $2$ $4$
280.192.4-280.c.2.23 $280$ $2$ $2$ $4$
280.192.4-280.d.1.25 $280$ $2$ $2$ $4$
280.192.4-280.d.1.26 $280$ $2$ $2$ $4$
280.192.4-280.d.2.25 $280$ $2$ $2$ $4$
280.192.4-280.d.2.27 $280$ $2$ $2$ $4$
280.192.5-280.y.1.24 $280$ $2$ $2$ $5$
280.192.5-280.bb.1.20 $280$ $2$ $2$ $5$
280.192.5-280.bk.1.22 $280$ $2$ $2$ $5$
280.192.5-280.bn.1.20 $280$ $2$ $2$ $5$
280.192.6-280.c.1.25 $280$ $2$ $2$ $6$
280.192.6-280.c.1.29 $280$ $2$ $2$ $6$
280.192.6-280.c.2.25 $280$ $2$ $2$ $6$
280.192.6-280.c.2.29 $280$ $2$ $2$ $6$
280.192.6-280.d.1.18 $280$ $2$ $2$ $6$
280.192.6-280.d.1.26 $280$ $2$ $2$ $6$
280.192.6-280.d.2.19 $280$ $2$ $2$ $6$
280.192.6-280.d.2.27 $280$ $2$ $2$ $6$