Modular curve 80.288.7-40.fn.2.22

• Label: 80.288.7-40.fn.2.22
• Level: $80$
• Index: $288$
• Genus: $7$
• Cusps: $12$
• $\Q$-cusps: $4$

Invariants

Level:	$80$		$\SL_2$-level:	$80$		Newform level:	$160$
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	$1^{2}\cdot2\cdot4\cdot5^{2}\cdot8^{2}\cdot10\cdot20\cdot40^{2}$		Cusp orbits	$1^{4}\cdot2^{4}$
Elliptic points:	$0$ of order $2$ and $0$ of order $3$
Analytic rank:	not computed
$\Q$-gonality:	$3 \le \gamma \le 7$
$\overline{\Q}$-gonality:	$3 \le \gamma \le 7$
Rational cusps:	$4$
Rational CM points:	none

Other labels

Cummins and Pauli (CP) label:

40V7

Level structure

$\GL_2(\Z/80\Z)$-generators:	$\begin{bmatrix}15&78\\4&49\end{bmatrix}$, $\begin{bmatrix}41&62\\30&33\end{bmatrix}$, $\begin{bmatrix}45&78\\56&67\end{bmatrix}$, $\begin{bmatrix}52&27\\13&26\end{bmatrix}$, $\begin{bmatrix}68&1\\21&8\end{bmatrix}$, $\begin{bmatrix}71&78\\52&57\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.144.7.fn.2 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 $	$=$	$ x t - x u + y t $
	$=$	$y t - y u - t v$
	$=$	$y w + y t - z v$
	$=$	$x w + x t + y z$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

$ x^{6} y^{2} + x^{6} z^{2} - 2 x^{5} y z^{2} - 2 x^{4} y^{4} + 3 x^{4} y^{2} z^{2} + 4 x^{3} y z^{4} + \cdots + y^{6} z^{2} $

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

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

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 -2^2\,\frac{191076449xzu^{9}v-11601927240xzu^{7}v^{3}-1838995115828xzu^{5}v^{5}-6711175082264xzu^{3}v^{7}-1065503194992xzuv^{9}-714136514xwu^{9}v-118527007532xwu^{7}v^{3}-4399622361584xwu^{5}v^{5}-740543716464xwu^{3}v^{7}+1861010640000xwuv^{9}-834195332xu^{10}v-139226771136xu^{8}v^{3}-2861744744592xu^{6}v^{5}-5208243273680xu^{4}v^{7}-8713005395008xu^{2}v^{9}-349644600000xv^{11}+1653269378yu^{10}v+125063882588yu^{8}v^{3}+2684501707168yu^{6}v^{5}+9704227092512yu^{4}v^{7}+7295891790464yu^{2}v^{9}+17558640000yv^{11}+299994847zwu^{10}+1229437428zwu^{8}v^{2}-933456871668zwu^{6}v^{4}-9478181610664zwu^{4}v^{6}-3003943459568zwu^{2}v^{8}+198533160000zwv^{10}+363994591zu^{11}-743459728zu^{9}v^{2}-796889449752zu^{7}v^{4}-9218531223712zu^{5}v^{6}-8878754881072zu^{3}v^{8}-1296418970016zuv^{10}-187946847wtu^{10}+4597051404wtu^{8}v^{2}-2071672793516wtu^{6}v^{4}-12992031831240wtu^{4}v^{6}-9356269700944wtu^{2}v^{8}-228073180032wtv^{10}+571940191wu^{11}-6057159017wu^{9}v^{2}+565988817496wu^{7}v^{4}-5183850005580wu^{5}v^{6}-6935508827864wu^{3}v^{8}-970939474992wuv^{10}-4000000t^{12}-8000000t^{10}v^{2}-32000000t^{8}v^{4}-116000000t^{6}v^{6}-454000000t^{4}v^{8}-1231886782t^{2}u^{10}+14514935828t^{2}u^{8}v^{2}-2540333510136t^{2}u^{6}v^{4}-1883742424992t^{2}u^{4}v^{6}+2713086968256t^{2}u^{2}v^{8}-1829000000t^{2}v^{10}+2131820317tu^{11}-18116161591tu^{9}v^{2}+1601489069904tu^{7}v^{4}-19984540751732tu^{5}v^{6}-23715602153576tu^{3}v^{8}-2943683553232tuv^{10}-575940191u^{12}+5733188009u^{10}v^{2}-561909736348u^{8}v^{4}+4898986448244u^{6}v^{6}+11111421513256u^{4}v^{8}+5730901324560u^{2}v^{10}+86685120000v^{12}}{5888xzu^{9}v+13923327xzu^{7}v^{3}+17295378xzu^{5}v^{5}+87820828xzu^{3}v^{7}+1145000xzuv^{9}+68608xwu^{9}v+35629058xwu^{7}v^{3}-112943912xwu^{5}v^{5}+78418248xwu^{3}v^{7}+36177600xwuv^{9}+69120xu^{10}v+43097092xu^{8}v^{3}-79325832xu^{6}v^{5}+250744544xu^{4}v^{7}-10557600xu^{2}v^{9}-7296000xv^{11}-63232yu^{10}v-27106562yu^{8}v^{3}+185299304yu^{6}v^{5}-210300344yu^{4}v^{7}-94443216yu^{2}v^{9}+368000yv^{11}-1729279zwu^{8}v^{2}-81451022zwu^{6}v^{4}+76256948zwu^{4}v^{6}+88220184zwu^{2}v^{8}+4144000zwv^{10}-2333439zu^{9}v^{2}-206950986zu^{7}v^{4}+296544160zu^{5}v^{6}+7326200zu^{3}v^{8}-25661200zuv^{10}-512wtu^{10}-10446081wtu^{8}v^{2}-358250738wtu^{6}v^{4}+282572748wtu^{4}v^{6}+80514008wtu^{2}v^{8}-5236000wtv^{10}+512wu^{11}+6315521wu^{9}v^{2}+38131823wu^{7}v^{4}+134586374wu^{5}v^{6}+3443004wu^{3}v^{8}-13637400wuv^{10}-1024t^{2}u^{10}-16236546t^{2}u^{8}v^{2}-319885664t^{2}u^{6}v^{4}+53209008t^{2}u^{4}v^{6}+63187264t^{2}u^{2}v^{8}-500000t^{2}v^{10}+1536tu^{11}+18421251tu^{9}v^{2}+37399433tu^{7}v^{4}+544601762tu^{5}v^{6}-7393276tu^{3}v^{8}-51853064tuv^{10}-512u^{12}-6315777u^{10}v^{2}-41157039u^{8}v^{4}-132772226u^{6}v^{6}-88860636u^{4}v^{8}+10877816u^{2}v^{10}+1808000v^{12}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 40.144.7.fn.2 :

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

Equation of the image curve:

$0$

$=$

$ X^{6}Y^{2}+X^{6}Z^{2}-2X^{5}YZ^{2}-2X^{4}Y^{4}+3X^{4}Y^{2}Z^{2}+4X^{3}YZ^{4}+X^{2}Y^{6}+3X^{2}Y^{4}Z^{2}+2XY^{5}Z^{2}-4XY^{3}Z^{4}+Y^{6}Z^{2} $

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.ba.1.8	$16$	$6$	$6$	$0$	$0$

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
16.48.0-8.ba.1.8	$16$	$6$	$6$	$0$	$0$
80.144.3-40.bx.1.18	$80$	$2$	$2$	$3$	$?$
80.144.3-40.bx.1.28	$80$	$2$	$2$	$3$	$?$