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Properties

Label 120.288.7-30.a.1.7
Level $120$
Index $288$
Genus $7$
Cusps $12$
$\Q$-cusps $12$

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Invariants

Level: $120$ $\SL_2$-level: $60$ Newform level: $60$
Index: $288$ $\PSL_2$-index:$144$
Genus: $7 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 12 }{2}$
Cusps: $12$ (all of which are rational) Cusp widths $2^{3}\cdot6^{3}\cdot10^{3}\cdot30^{3}$ Cusp orbits $1^{12}$
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: $12$
Rational CM points: none

Other labels

Cummins and Pauli (CP) label: 30Q7

Level structure

$\GL_2(\Z/120\Z)$-generators: $\begin{bmatrix}19&80\\0&107\end{bmatrix}$, $\begin{bmatrix}29&4\\110&21\end{bmatrix}$, $\begin{bmatrix}47&12\\30&71\end{bmatrix}$, $\begin{bmatrix}47&96\\90&101\end{bmatrix}$, $\begin{bmatrix}63&110\\70&77\end{bmatrix}$, $\begin{bmatrix}83&22\\90&43\end{bmatrix}$, $\begin{bmatrix}89&36\\50&7\end{bmatrix}$, $\begin{bmatrix}99&38\\100&107\end{bmatrix}$
Contains $-I$: no $\quad$ (see 30.144.7.a.1 for the level structure with $-I$)
Cyclic 120-isogeny field degree: $4$
Cyclic 120-torsion field degree: $128$
Full 120-torsion field degree: $122880$

Models

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

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

$ 0 $ $=$ $ - 9 x^{4} y^{2} z^{2} + 9 x^{4} z^{4} + x^{2} y^{6} + 2 x^{2} y^{5} z - 4 x^{2} y^{4} z^{2} + \cdots + y^{4} z^{4} $
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Rational points

This modular curve has 12 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.

Canonical model
$(0:0:1:0:0:0:0)$, $(-1/2:0:-3/2:-3/2:-1/2:1:1)$, $(0:0:0:1:0:0:0)$, $(0:0:-1:-1:-1:1:1)$, $(0:0:0:0:0:0:1)$, $(1:0:0:0:1:1:1)$, $(-1/2:0:0:0:1:-1/2:1)$, $(0:0:0:0:1:0:0)$, $(1:0:0:0:-2:-2:1)$, $(1:0:-3:0:-2:1:1)$, $(0:0:0:0:0:1:0)$, $(1:0:0:-3:-2:1:1)$

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{9501687125660xu^{11}+24010109290355xu^{10}v-435665382644693xu^{9}v^{2}+1321423945517942xu^{8}v^{3}-1525978728995808xu^{7}v^{4}-80908571319164xu^{6}v^{5}+2123261559811288xu^{5}v^{6}-2249238291731124xu^{4}v^{7}+732690262923156xu^{3}v^{8}+294233312638608xu^{2}v^{9}-245609678668248xuv^{10}+42793401615648xv^{11}+697494663388yu^{11}+17307345823291yu^{10}v-134072221137913yu^{9}v^{2}+359582163713284yu^{8}v^{3}-397292374892208yu^{7}v^{4}-69417423608956yu^{6}v^{5}+709461050314000yu^{5}v^{6}-807998842845116yu^{4}v^{7}+321556684023956yu^{3}v^{8}+94665195428200yu^{2}v^{9}-122114119975256yuv^{10}+23651350962080yv^{11}+8000000z^{12}+96000000z^{11}v+576000000z^{10}v^{2}+2432000000z^{9}v^{3}+8496000000z^{8}v^{4}+26784000000z^{7}v^{5}+79840000000z^{6}v^{6}+230592000000z^{5}v^{7}+652488000000z^{4}v^{8}+1817600000000z^{3}v^{9}+4995456000000z^{2}v^{10}-4033060689952ztu^{10}+7683721331570ztu^{9}v+82623296238636ztu^{8}v^{2}-308219959308024ztu^{7}v^{3}+351206808262760ztu^{6}v^{4}+42729167002448ztu^{5}v^{5}-447316212505200ztu^{4}v^{6}+365416439138856ztu^{3}v^{7}-33991358472040ztu^{2}v^{8}-73772075984216ztuv^{9}+20919313453912ztv^{10}-7557712087006zu^{11}-23104293651950zu^{10}v+404042917242290zu^{9}v^{2}-1282280645349678zu^{8}v^{3}+1577049653100928zu^{7}v^{4}-25007033797744zu^{6}v^{5}-2189599949382328zu^{5}v^{6}+2534260281655680zu^{4}v^{7}-966027300750360zu^{3}v^{8}-284079319451448zu^{2}v^{9}+330793893635984zuv^{10}-54933179164368zv^{11}+8000000w^{12}-48000000w^{10}v^{2}+24000000w^{8}v^{4}-192000000w^{7}v^{5}-208000000w^{6}v^{6}-192000000w^{5}v^{7}+264000000w^{4}v^{8}+7200000000w^{3}v^{9}+51936000000w^{2}v^{10}-3478298392681wtu^{10}-17532155276805wtu^{9}v+183736258104566wtu^{8}v^{2}-423806810471716wtu^{7}v^{3}+276350031768960wtu^{6}v^{4}+344554602484592wtu^{5}v^{5}-722149588661616wtu^{4}v^{6}+426841248160596wtu^{3}v^{7}+25872002406536wtu^{2}v^{8}-121627291609840wtuv^{9}+24813273101208wtv^{10}+7974028919wu^{11}-19051710426223wu^{10}v+59536837152334wu^{9}v^{2}+63808631894662wu^{8}v^{3}-418200395321504wu^{7}v^{4}+497495299468872wu^{6}v^{5}+49054608136384wu^{5}v^{6}-589583724482344wu^{4}v^{7}+464841319925872wu^{3}v^{8}-35394584075144wu^{2}v^{9}-109842815112936wuv^{10}+25115193101208wv^{11}+512t^{12}-3072t^{10}v^{2}+1536t^{8}v^{4}+24576t^{7}v^{5}+281600t^{6}v^{6}-1867776t^{5}v^{7}+8434176t^{4}v^{8}-40605696t^{3}v^{9}+1232123331104t^{2}u^{10}-256505145005t^{2}u^{9}v-29563587343592t^{2}u^{8}v^{2}+71181452702292t^{2}u^{7}v^{3}-37871746608688t^{2}u^{6}v^{4}-58317103087264t^{2}u^{5}v^{5}+87913171218888t^{2}u^{4}v^{6}-27720640406836t^{2}u^{3}v^{7}-17938614779888t^{2}u^{2}v^{8}+5587328292960t^{2}uv^{9}+191404032t^{2}v^{10}+2776170675639tu^{11}-2923794658928tu^{10}v-103916645500775tu^{9}v^{2}+480087475715443tu^{8}v^{3}-820276492878252tu^{7}v^{4}+368633408505556tu^{6}v^{5}+746395466972680tu^{5}v^{6}-1258155199798660tu^{4}v^{7}+666438819766204tu^{3}v^{8}+61030292413612tu^{2}v^{9}-185897773098440tuv^{10}+40144375449232tv^{11}+512u^{12}-697494663388u^{11}v-19948815522261u^{10}v^{2}+112565542962311u^{9}v^{3}-180696758498224u^{8}v^{4}+24850433525992u^{7}v^{5}+264145872671868u^{6}v^{6}-337989113542344u^{5}v^{7}+140126428337092u^{4}v^{8}+43696831668308u^{3}v^{9}-53765028525224u^{2}v^{10}+8622168614056uv^{11}+512v^{12}}{6319452xu^{11}-6917315xu^{10}v-11108035xu^{9}v^{2}-64052028xu^{8}v^{3}+308121662xu^{7}v^{4}-491997806xu^{6}v^{5}+437435898xu^{5}v^{6}-283456824xu^{4}v^{7}+154276660xu^{3}v^{8}-61879888xu^{2}v^{9}+13006368xuv^{10}-445296xv^{11}+650564yu^{11}+501729yu^{10}v-2642247yu^{9}v^{2}-24522252yu^{8}v^{3}+111590882yu^{7}v^{4}-204689310yu^{6}v^{5}+217042106yu^{5}v^{6}-159494372yu^{4}v^{7}+92254660yu^{3}v^{8}-40968752yu^{2}v^{9}+12088064yuv^{10}-1811072yv^{11}-2389098ztu^{10}+2952007ztu^{9}v+8924643ztu^{8}v^{2}-4236688ztu^{7}v^{3}-32410088ztu^{6}v^{4}+43259512ztu^{5}v^{5}-23762864ztu^{4}v^{6}+15065796ztu^{3}v^{7}-9866420ztu^{2}v^{8}+1967904ztuv^{9}+495296ztv^{10}-5090714zu^{11}+5731526zu^{10}v+13622422zu^{9}v^{2}+46684194zu^{8}v^{3}-282068344zu^{7}v^{4}+490774684zu^{6}v^{5}-458425548zu^{5}v^{6}+298310164zu^{4}v^{7}-164413048zu^{3}v^{8}+73409224zu^{2}v^{9}-21708240zuv^{10}+3173680zv^{11}-2353913wtu^{10}-813848wtu^{9}v+9040525wtu^{8}v^{2}+25703846wtu^{7}v^{3}-95938364wtu^{6}v^{4}+115114262wtu^{5}v^{5}-82562488wtu^{4}v^{6}+50173400wtu^{3}v^{7}-25021892wtu^{2}v^{8}+8155488wtuv^{9}-1390648wtv^{10}-143411wu^{11}-4443119wu^{10}v+1194021wu^{9}v^{2}+25109393wu^{8}v^{3}-17731664wu^{7}v^{4}-45948984wu^{6}v^{5}+79148164wu^{5}v^{6}-61241400wu^{4}v^{7}+39535604wu^{3}v^{8}-20995196wu^{2}v^{9}+7119976wuv^{10}-1390648wv^{11}-512t^{6}v^{6}+3072t^{5}v^{7}-13824t^{4}v^{8}+65536t^{3}v^{9}+719658t^{2}u^{10}+286526t^{2}u^{9}v-4807944t^{2}u^{8}v^{2}+3793330t^{2}u^{7}v^{3}-2382100t^{2}u^{6}v^{4}+8338666t^{2}u^{5}v^{5}-8118256t^{2}u^{4}v^{6}+3437536t^{2}u^{3}v^{7}-1991432t^{2}u^{2}v^{8}+1566736t^{2}uv^{9}-305664t^{2}v^{10}+1810013tu^{11}-5469124tu^{10}v+579817tu^{9}v^{2}-12530128tu^{8}v^{3}+107061522tu^{7}v^{4}-228787464tu^{6}v^{5}+240545430tu^{5}v^{6}-164506566tu^{4}v^{7}+94145284tu^{3}v^{8}-43045288tu^{2}v^{9}+11484544tuv^{10}-1061256tv^{11}-650564u^{11}v-2381031u^{10}v^{2}+1325213u^{9}v^{3}+29017196u^{8}v^{4}-72293114u^{7}v^{5}+87100698u^{6}v^{6}-71803030u^{5}v^{7}+44754068u^{4}v^{8}-20855516u^{3}v^{9}+7805584u^{2}v^{10}-2140464uv^{11}}$

Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 30.144.7.a.1 :

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

Equation of the image curve:

$0$ $=$ $ -9X^{4}Y^{2}Z^{2}+9X^{4}Z^{4}+X^{2}Y^{6}+2X^{2}Y^{5}Z-4X^{2}Y^{4}Z^{2}+4X^{2}Y^{2}Z^{4}+2X^{2}YZ^{5}-X^{2}Z^{6}-Y^{6}Z^{2}+Y^{4}Z^{4} $

Modular covers

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

Factor curve Level Index Degree Genus Rank
$X_0(3)$ $3$ $72$ $36$ $0$ $0$
40.72.1-10.a.1.1 $40$ $4$ $4$ $1$ $0$

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank
40.72.1-10.a.1.1 $40$ $4$ $4$ $1$ $0$
120.48.0-6.a.1.7 $120$ $6$ $6$ $0$ $?$