Invariants
Level: | $280$ | $\SL_2$-level: | $56$ | Newform level: | $448$ | ||
Index: | $192$ | $\PSL_2$-index: | $96$ | ||||
Genus: | $6 = 1 + \frac{ 96 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $2^{2}\cdot8\cdot14^{2}\cdot56$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
Analytic rank: | not computed | ||||||
$\Q$-gonality: | $3 \le \gamma \le 6$ | ||||||
$\overline{\Q}$-gonality: | $3 \le \gamma \le 6$ | ||||||
Rational cusps: | $2$ | ||||||
Rational CM points: | none |
Other labels
Cummins and Pauli (CP) label: | 56D6 |
Level structure
$\GL_2(\Z/280\Z)$-generators: | $\begin{bmatrix}17&152\\124&31\end{bmatrix}$, $\begin{bmatrix}95&276\\176&247\end{bmatrix}$, $\begin{bmatrix}157&216\\232&75\end{bmatrix}$, $\begin{bmatrix}214&189\\237&136\end{bmatrix}$, $\begin{bmatrix}242&13\\223&88\end{bmatrix}$ |
Contains $-I$: | no $\quad$ (see 56.96.6.d.2 for the level structure with $-I$) |
Cyclic 280-isogeny field degree: | $24$ |
Cyclic 280-torsion field degree: | $2304$ |
Full 280-torsion field degree: | $7741440$ |
Models
Canonical model in $\mathbb{P}^{ 5 }$ defined by 9 equations
$ 0 $ | $=$ | $ y w - z t $ |
$=$ | $x w + y z$ | |
$=$ | $x t + y^{2}$ | |
$=$ | $3 x y - y u + w t$ | |
$=$ | $\cdots$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 7 x^{8} + 7 x^{4} y^{2} z^{2} - 9 x^{4} y z^{3} + x^{4} z^{4} - y^{3} z^{5} + y^{2} z^{6} $ |
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.
Canonical model |
---|
$(0:0:0:0:0:1)$, $(0:0:1:0:0:0)$ |
Maps to other modular curves
$j$-invariant map of degree 96 from the canonical model of this modular curve to the modular curve $X(1)$ :
$\displaystyle j$ | $=$ | $\displaystyle -\frac{3}{2^2}\cdot\frac{21364633280xt^{8}u+45469881216xt^{6}u^{3}-101042464436xt^{4}u^{5}-341836293884xt^{2}u^{7}-91663474686xu^{9}+4586471424z^{10}-6306398208z^{8}u^{2}+15497256960z^{6}u^{4}-9066147264z^{4}u^{6}+12749164614z^{2}u^{8}-1119465664zt^{9}-19179658928zt^{7}u^{2}+200184081644zt^{5}u^{4}-246756019589zt^{3}u^{6}+84332742194ztu^{8}+14930313728w^{2}t^{7}u-23518551376w^{2}t^{5}u^{3}-118462401528w^{2}t^{3}u^{5}-119913697559w^{2}tu^{7}-56405568t^{10}+4994113200t^{8}u^{2}-72292394124t^{6}u^{4}+74549340453t^{4}u^{6}+188010715809t^{2}u^{8}+32105299968u^{10}}{55552xt^{8}u+616056xt^{6}u^{3}+5246900xt^{4}u^{5}+4099733xt^{2}u^{7}-49707xu^{9}-13436928z^{6}u^{4}+9237888z^{4}u^{6}+1043199z^{2}u^{8}-152096zt^{9}-455896zt^{7}u^{2}-226610zt^{5}u^{4}-2719417zt^{3}u^{6}+776845ztu^{8}-736736w^{2}t^{7}u-638624w^{2}t^{5}u^{3}+3351342w^{2}t^{3}u^{5}-610108w^{2}tu^{7}+479136t^{10}+1576344t^{8}u^{2}+869058t^{6}u^{4}-1487793t^{4}u^{6}-1259643t^{2}u^{8}}$ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 56.96.6.d.2 :
$\displaystyle X$ | $=$ | $\displaystyle y$ |
$\displaystyle Y$ | $=$ | $\displaystyle z$ |
$\displaystyle Z$ | $=$ | $\displaystyle t$ |
Equation of the image curve:
$0$ | $=$ | $ 7X^{8}+7X^{4}Y^{2}Z^{2}-9X^{4}YZ^{3}+X^{4}Z^{4}-Y^{3}Z^{5}+Y^{2}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
Covered curve | Level | Index | Degree | Genus | Rank |
---|---|---|---|---|---|
280.96.2-28.b.1.1 | $280$ | $2$ | $2$ | $2$ | $?$ |
280.96.2-28.b.1.15 | $280$ | $2$ | $2$ | $2$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.
Covering curve | Level | Index | Degree | Genus |
---|---|---|---|---|
280.384.11-56.l.1.11 | $280$ | $2$ | $2$ | $11$ |
280.384.11-56.bu.2.2 | $280$ | $2$ | $2$ | $11$ |
280.384.11-56.cj.1.6 | $280$ | $2$ | $2$ | $11$ |
280.384.11-56.co.1.7 | $280$ | $2$ | $2$ | $11$ |
280.384.11-56.ed.2.7 | $280$ | $2$ | $2$ | $11$ |
280.384.11-56.ef.2.1 | $280$ | $2$ | $2$ | $11$ |
280.384.11-56.eh.2.5 | $280$ | $2$ | $2$ | $11$ |
280.384.11-56.ej.2.5 | $280$ | $2$ | $2$ | $11$ |
280.384.11-280.ip.1.14 | $280$ | $2$ | $2$ | $11$ |
280.384.11-280.ir.2.15 | $280$ | $2$ | $2$ | $11$ |
280.384.11-280.it.1.11 | $280$ | $2$ | $2$ | $11$ |
280.384.11-280.iv.1.12 | $280$ | $2$ | $2$ | $11$ |
280.384.11-280.mh.2.14 | $280$ | $2$ | $2$ | $11$ |
280.384.11-280.mj.2.13 | $280$ | $2$ | $2$ | $11$ |
280.384.11-280.ml.2.9 | $280$ | $2$ | $2$ | $11$ |
280.384.11-280.mn.2.11 | $280$ | $2$ | $2$ | $11$ |