Modular curve 80.288.7-40.eg.1.10

• Label: 80.288.7-40.eg.1.10
• Level: $80$
• Index: $288$
• Genus: $7$
• Cusps: $12$
• $\Q$-cusps: $4$

Invariants

Level:	$80$		$\SL_2$-level:	$80$		Newform level:	$400$
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:	$\begin{bmatrix}21&60\\78&63\end{bmatrix}$, $\begin{bmatrix}23&40\\31&13\end{bmatrix}$, $\begin{bmatrix}39&40\\24&31\end{bmatrix}$, $\begin{bmatrix}59&40\\45&21\end{bmatrix}$, $\begin{bmatrix}69&40\\31&39\end{bmatrix}$, $\begin{bmatrix}79&40\\21&1\end{bmatrix}$
Contains $-I$:	no $\quad$ (see 40.144.7.eg.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 $	$=$	$ x^{2} - y u $
	$=$	$x z - x v + 2 z u + t u$
	$=$	$2 x z - x t - w u$
	$=$	$2 x z + x t + y z - y v$
	$=$	$\cdots$

Singular plane model Singular plane model

$ 0 $

$=$

$ 625 x^{8} - 125 x^{6} y^{2} + 500 x^{6} z^{2} + 25 x^{4} y^{2} z^{2} + 150 x^{4} z^{4} + 5 x^{2} y^{4} z^{2} + \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

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

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}{3^{10}\cdot5^5}\cdot\frac{3499361707276859062500000xu^{11}-9562472990855289562500000xu^{9}v^{2}+16096319162532028743750000xu^{7}v^{4}-21113039792194402405950000xu^{5}v^{6}+15382898690345232665298000xu^{3}v^{8}-3321125739401305509487680xuv^{10}-208467515376919687500000ytu^{9}v+1837343699682226425000000ytu^{7}v^{3}-2323090227470098050900000ytu^{5}v^{5}+1887332628192855937506000ytu^{3}v^{7}-544450675077121918974560ytuv^{9}+3818330605090572451171875yu^{11}-2343773391219192005859375yu^{9}v^{2}+6137100053744529121875000yu^{7}v^{4}-4127828877166170096337500yu^{5}v^{6}+539298077559734863800375yu^{3}v^{8}+778274030129132332383680yuv^{10}+16233255069613204078125000zu^{10}v-22615125443820822740625000zu^{8}v^{3}+33500350863265676739300000zu^{6}v^{5}-27565262989774180982802000zu^{4}v^{7}+7066333225664807776279720zu^{2}v^{9}-179137996098426322744704zv^{11}-9226406250000w^{12}-110716875000000w^{11}v+110716875000000w^{10}tv-276792187500000w^{10}v^{2}+332150625000000w^{9}tv^{2}-535131562500000w^{9}v^{3}+1826828437500000w^{8}tv^{3}+3847411406250000w^{8}v^{4}-15315834375000000w^{7}tv^{4}-53513156250000000w^{7}v^{5}+230752420312500000w^{6}tv^{5}+795491520468750000w^{6}v^{6}-3435212480625000000w^{5}tv^{6}-11999374969218750000w^{5}v^{7}+52481062767656250000w^{4}tv^{7}+185809303771875000000w^{4}v^{8}-820637684741250000000w^{3}tv^{8}-2936740760888906250000w^{3}v^{9}+13074636536710312500000w^{2}tv^{9}+47203999848516093750000w^{2}v^{10}-5582220274988886467648wtv^{10}+2310391905504434437500000wu^{10}v-3082851322949602012500000wu^{8}v^{3}+4787216663867649321300000wu^{6}v^{5}-3847833069338562836742000wu^{4}v^{7}+793788929294502939871920wu^{2}v^{9}+11164440574084727064704wv^{11}+275176359284701552734375t^{2}u^{10}+859102575019486611328125t^{2}u^{8}v^{2}-339743425209902173125000t^{2}u^{6}v^{4}+381010638267773257792500t^{2}u^{4}v^{6}-227404129033450938608325t^{2}u^{2}v^{8}-591779700736t^{2}v^{10}+9313868699785773562500000tu^{10}v-11268430948163647462500000tu^{8}v^{3}+18135027411984498636450000tu^{6}v^{5}-14936742760235947458768000tu^{4}v^{7}+4017463079204154947323680tu^{2}v^{9}+89568998049121875000000tv^{11}+488389749661917773437500u^{12}-1472315573904062062500000u^{10}v^{2}+1993149889063663514062500u^{8}v^{4}-1990322631985427942400000u^{6}v^{6}+913393520053763302033500u^{4}v^{8}+280748798041475574190940u^{2}v^{10}+131044507648v^{12}}{u(7588312550000xu^{10}-6714983460000xu^{8}v^{2}-2495126458000xu^{6}v^{4}+218327051200xu^{4}v^{6}+8471587840xu^{2}v^{8}+15892480xv^{10}+1220291300000ytu^{8}v+509173300000ytu^{6}v^{3}+58957353600ytu^{4}v^{5}+1196904320ytu^{2}v^{7}+2007040ytv^{9}+5642108215625yu^{10}+8775673490625yu^{8}v^{2}+795941422500yu^{6}v^{4}-74404995500yu^{4}v^{6}-2280063040yu^{2}v^{8}-4009984yv^{10}+27350019795000zu^{9}v+3316607231000zu^{7}v^{3}-643074259200zu^{5}v^{5}-18580263200zu^{3}v^{7}-31933952zuv^{9}+4070567000000wu^{9}v+860969360000wu^{7}v^{3}-51433319200wu^{5}v^{5}-2114471680wu^{3}v^{7}-3973120wuv^{9}+1444955453125t^{2}u^{9}+1809173728125t^{2}u^{7}v^{2}+178513162500t^{2}u^{5}v^{4}+2363008500t^{2}u^{3}v^{6}+2577600t^{2}uv^{8}+16593325590000tu^{9}v+4852159612000tu^{7}v^{3}-74682154000tu^{5}v^{5}-7134790720tu^{3}v^{7}-15151104tuv^{9}+104270312500u^{11}-493469165000u^{9}v^{2}-217256899500u^{7}v^{4}-25202479200u^{5}v^{6}-468058160u^{3}v^{8}-650496uv^{10})}$

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

$\displaystyle X$	$=$	$\displaystyle y$
$\displaystyle Y$	$=$	$\displaystyle 4z$
$\displaystyle Z$	$=$	$\displaystyle w$

Equation of the image curve:

$0$

$=$

$ 625X^{8}-125X^{6}Y^{2}+500X^{6}Z^{2}+25X^{4}Y^{2}Z^{2}+150X^{4}Z^{4}+5X^{2}Y^{4}Z^{2}+5X^{2}Y^{2}Z^{4}+20X^{2}Z^{6}-Y^{2}Z^{6}+Z^{8} $

Modular covers

This modular curve minimally covers the modular curves listed below.

Covered curve	Level	Index	Degree	Genus	Rank
80.48.0-40.bn.1.7	$80$	$6$	$6$	$0$	$?$
80.144.3-40.bx.1.10	$80$	$2$	$2$	$3$	$?$
80.144.3-40.bx.1.18	$80$	$2$	$2$	$3$	$?$