Modular curve 80.288.7-40.cm.1.12

• Label: 80.288.7-40.cm.1.12
• Level: $80$
• Index: $288$
• Genus: $7$
• Cusps: $12$
• $\Q$-cusps: $4$

Invariants

Level:	$80$		$\SL_2$-level:	$80$		Newform level:	$320$
Index:	$288$		$\PSL_2$-index:	$144$
Genus:	$7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps:	$12$ (of which $4$ are rational)		Cusp widths	$2^{4}\cdot8^{2}\cdot10^{4}\cdot40^{2}$		Cusp orbits	$1^{4}\cdot2^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$4 \le \gamma \le 7$
$\overline{\Q}$-gonality:	$4 \le \gamma \le 7$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

40M7

Level structure

$\GL_2(\Z/80\Z)$-generators:	$\begin{bmatrix}21&20\\39&69\end{bmatrix}$, $\begin{bmatrix}27&60\\8&69\end{bmatrix}$, $\begin{bmatrix}33&20\\19&17\end{bmatrix}$, $\begin{bmatrix}57&60\\69&41\end{bmatrix}$, $\begin{bmatrix}69&20\\75&53\end{bmatrix}$, $\begin{bmatrix}69&40\\27&23\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.144.7.cm.1 for the level structure with $-I$)
Cyclic 80-isogeny field degree:	$2$
Cyclic 80-torsion field degree:	$32$
Full 80-torsion field degree:	$40960$

Models

Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations

$ 0 $	$=$	$ y^{2} - z u $
	$=$	$ - x w + y^{2} + z u + w v$
	$=$	$x y - 2 x u - y v + t u$
	$=$	$2 x y + y t + w u$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

$ - 2 x^{4} y^{2} z^{2} + 8 x^{2} y^{6} + 4 x^{2} y^{4} z^{2} - 2 x^{2} y^{2} z^{4} - x^{2} z^{6} + \cdots + z^{8} $

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.

Canonical model

$(1:0:0:-4:2:0:1)$, $(1/5:0:0:0:-2/5:0:1)$, $(-1/3:0:0:0:2/3:0:1)$, $(1:0:0:4:2:0:1)$

Maps to other modular curves

$j$-invariant map of degree 144 from the canonical model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$

$=$

$\displaystyle \frac{1}{2^3\cdot3^{10}\cdot5^4}\cdot\frac{83114265956419604880000xu^{10}v+289473605680906531080000xu^{8}v^{3}+1072011227624501655657600xu^{6}v^{5}+2205221039181934478624160xu^{4}v^{7}+1413266645132961555255944xu^{2}v^{9}+89568998049213161372352xv^{11}+7166692776503007360000yu^{11}+48959861713179082560000yu^{9}v^{2}+206032885280409967920000yu^{7}v^{4}+675617273350220876990400yu^{5}v^{6}+1230631895227618613223840yu^{3}v^{8}+664225147880261101897536yuv^{10}+1067353678729828800000ztu^{9}v+23517999355932498240000ztu^{7}v^{3}+74338887279043137628800ztu^{5}v^{5}+150986610255428475000480ztu^{3}v^{7}+108890135015424383794912ztuv^{9}-7819941079225492380000zu^{11}-12000119763042263070000zu^{9}v^{2}-78554880687929972760000zu^{7}v^{4}-132090524069317443082800zu^{5}v^{6}-43143846204778789104030zu^{3}v^{8}+155654806025826466476736zuv^{10}+4613203125000w^{12}+55358437500000w^{11}v+55358437500000w^{10}tv+138396093750000w^{10}v^{2}+166075312500000w^{9}tv^{2}+267565781250000w^{9}v^{3}+913414218750000w^{8}tv^{3}-1923705703125000w^{8}v^{4}-7657917187500000w^{7}tv^{4}+26756578125000000w^{7}v^{5}+115376210156250000w^{6}tv^{5}-397745760234375000w^{6}v^{6}-1717606240312500000w^{5}tv^{6}+5999687484609375000w^{5}v^{7}+26240531383828125000w^{4}tv^{7}-92904651885937500000w^{4}v^{8}-410318842370625000000w^{3}tv^{8}+1468370380444453125000w^{3}v^{9}+6537318268355156250000w^{2}tv^{9}-23601999924258046875000w^{2}v^{10}-2791110137494443233824wtv^{10}+11829206556182704320000wu^{10}v+39460496933754905760000wu^{8}v^{3}+153190933243764778281600wu^{6}v^{5}+307826645547085026939360wu^{4}v^{7}+158757785858900587974384wu^{2}v^{9}-5582220287042363532352wv^{11}+1408902959537671950000t^{2}u^{10}-10996512960249428625000t^{2}u^{8}v^{2}-10871789606716869540000t^{2}u^{6}v^{4}-30480851061421860623400t^{2}u^{4}v^{6}-45480825806690187721665t^{2}u^{2}v^{8}+295889850368t^{2}v^{10}-47687007742903160640000tu^{10}v-144235916136494687520000tu^{8}v^{3}-580320877183503956366400tu^{6}v^{5}-1194939420818875796701440tu^{4}v^{7}-803492615840830989464736tu^{2}v^{9}+44784499024560937500000tv^{11}-1000222207307607600000u^{12}-7538255738388797760000u^{10}v^{2}-25512318580014892980000u^{8}v^{4}-63690324223533694156800u^{6}v^{6}-73071481604301064162680u^{4}v^{8}+56149759608295114838188u^{2}v^{10}-65522253824v^{12}}{u(87520063344xu^{9}v-26532857848xu^{7}v^{3}-12861485184xu^{5}v^{5}+929013160xu^{3}v^{7}-3991744xuv^{9}+9713040064yu^{10}+21487947072yu^{8}v^{2}-19961011664yu^{6}v^{4}-4366541024yu^{4}v^{6}+423579392yu^{2}v^{8}-1986560yv^{10}-3904932160ztu^{8}v+4073386400ztu^{6}v^{3}-1179147072ztu^{4}v^{5}+59845216ztu^{2}v^{7}-250880ztv^{9}-7221898516zu^{10}+28082155170zu^{8}v^{2}-6367531380zu^{6}v^{4}-1488099910zu^{4}v^{6}+114003152zu^{2}v^{8}-501248zv^{10}+13025814400wu^{9}v-6887754880wu^{7}v^{3}-1028666384wu^{5}v^{5}+105723584wu^{3}v^{7}-496640wuv^{9}+4623857450t^{2}u^{9}-14473389825t^{2}u^{7}v^{2}+3570263250t^{2}u^{5}v^{4}-118150425t^{2}u^{3}v^{6}+322200t^{2}uv^{8}-53098641888tu^{9}v+38817276896tu^{7}v^{3}+1493643080tu^{5}v^{5}-356739536tu^{3}v^{7}+1893888tuv^{9}-133466000u^{11}-1579101328u^{9}v^{2}+1738055196u^{7}v^{4}-504049584u^{5}v^{6}+23402908u^{3}v^{8}-81312uv^{10})}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.144.7.cm.1 :

$\displaystyle X$	$=$	$\displaystyle x$
$\displaystyle Y$	$=$	$\displaystyle \frac{1}{4}z$
$\displaystyle Z$	$=$	$\displaystyle \frac{1}{4}w$

Equation of the image curve:

$0$

$=$

$ -2X^{4}Y^{2}Z^{2}+8X^{2}Y^{6}+4X^{2}Y^{4}Z^{2}-2X^{2}Y^{2}Z^{4}-X^{2}Z^{6}+16Y^{8}-32Y^{6}Z^{2}+24Y^{4}Z^{4}-8Y^{2}Z^{6}+Z^{8} $

Modular covers

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

Factor curve	Level	Index	Degree	Genus	Rank
$X_0(5)$	$5$	$48$	$24$	$0$	$0$
16.48.0-8.r.1.3	$16$	$6$	$6$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
16.48.0-8.r.1.3	$16$	$6$	$6$	$0$	$0$
80.144.3-40.bx.1.16	$80$	$2$	$2$	$3$	$?$
80.144.3-40.bx.1.18	$80$	$2$	$2$	$3$	$?$