Embedded model Embedded model in $\mathbb{P}^{6}$
| $ 0 $ | $=$ | $ z^{2} v + z w v + z t v + w^{2} v - w u v + t u v $ |
| $=$ | $z^{2} u + z w u + z t u + w^{2} u - w u^{2} + t u^{2}$ |
| $=$ | $x z v - y z v - z^{2} v + w t v + w u v - u^{2} v$ |
| $=$ | $x z v - y z v + z^{2} v - z t v - w^{2} v - w u v$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 11 x^{7} + 53 x^{6} z + 615 x^{5} y^{2} + 81 x^{5} z^{2} + 1175 x^{4} y^{2} z + 40 x^{4} z^{3} + \cdots - 11 z^{7} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -x^{11} + 11x^{6} + x $ |
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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.
Maps to other modular curves
$j$-invariant map
of degree 120 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{1}{7\cdot241^9}\cdot\frac{902296517404691571629369xy^{10}+14110381708349963939310345xy^{8}v^{2}+277024228641482965502378610xy^{6}v^{4}+7195238792355135693379347965xy^{4}v^{6}+220496407704982384455467570135xy^{2}v^{8}+154801535308017757442960432xu^{10}+358348276444859997095903593xu^{8}v^{2}+13395740848811990913659222339xu^{6}v^{4}+405051707666348538477320287579xu^{4}v^{6}+3599606475954634359572472751455xu^{2}v^{8}+7251108819932650011617529032537xv^{10}+2361329183846320495966221y^{11}+14686315655629554304180155y^{9}v^{2}+121330084893567036865906640y^{7}v^{4}+2711305045809884907685442210y^{5}v^{6}+82786569372801371082541910465y^{3}v^{8}+414497995186353281009055438yu^{10}+894645860210280385470259402yu^{8}v^{2}+33001942638057868915586144166yu^{6}v^{4}+949262145585904209228315301071yu^{4}v^{6}+7033953751006684066552229827795yu^{2}v^{8}+2725338713116457230188151896308yv^{10}-64022421860946988117002282ztu^{9}-23509807524452153333702966ztu^{7}v^{2}-11267497952270092510567151702ztu^{5}v^{4}-268585639897945724214248841831ztu^{3}v^{6}-1972331168698544044301417112620ztuv^{8}-100151038536538982739855172zu^{10}-684447021806665796076410085zu^{8}v^{2}-5253153630691605780838706136zu^{6}v^{4}-203946771382244874624066033767zu^{4}v^{6}-1565876792291232862020557199515zu^{2}v^{8}-1727620560328121224474828174125zv^{10}-123711631507311139939571501wtu^{9}+241371023290596053352392158wtu^{7}v^{2}-5013944902672155496250286750wtu^{5}v^{4}-144813037501326288555622383769wtu^{3}v^{6}-948402972472209517522166961855wtuv^{8}+196007284779186446733000407wu^{10}+75400110819339202632374506wu^{8}v^{2}+8886167536576766861677509012wu^{6}v^{4}+276473525147366418834453310456wu^{4}v^{6}+2135949544412816775032055307670wu^{2}v^{8}+1422899087305363447837955121000wv^{10}+14265603711372162303255485t^{2}u^{9}+346948092775833030399608296t^{2}u^{7}v^{2}+2217582406101582542338582251t^{2}u^{5}v^{4}+74675784490893812182860236979t^{2}u^{3}v^{6}+622179860450123850992896897105t^{2}uv^{8}+164810862634875301969219032tu^{10}-433270895856741131317939244tu^{8}v^{2}+11197612233728049511266759497tu^{6}v^{4}+279335323303497135110774754696tu^{4}v^{6}+2251873317136047678734105482695tu^{2}v^{8}+2146261233075195511463002794000tv^{10}+156537631223946983292792408u^{11}+713856737585791753497557183u^{9}v^{2}+13950976735189158934618268692u^{7}v^{4}+406844563860221118278266450510u^{5}v^{6}+2763660036195128578089113262490u^{3}v^{8}-949044898560143045994500581575uv^{10}}{v^{10}(3x+2y+4z-11w-8t+3u)}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
20.120.5.p.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle v$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 11X^{7}+615X^{5}Y^{2}+53X^{6}Z+1175X^{4}Y^{2}Z+81X^{5}Z^{2}+900X^{3}Y^{2}Z^{2}+40X^{4}Z^{3}+350X^{2}Y^{2}Z^{3}-40X^{3}Z^{4}+75XY^{2}Z^{4}-81X^{2}Z^{5}+10Y^{2}Z^{5}-53XZ^{6}-11Z^{7} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
20.120.5.p.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{3}{5}x-\frac{2}{5}y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{123}{625}x^{5}v-\frac{47}{125}x^{4}yv-\frac{36}{125}x^{3}y^{2}v-\frac{14}{125}x^{2}y^{3}v-\frac{3}{125}xy^{4}v-\frac{2}{625}y^{5}v$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{1}{5}x+\frac{1}{5}y$ |
The following modular covers realize this modular curve as a fiber product over $X(1)$.
This modular curve minimally covers the modular curves listed below.
This modular curve is minimally covered by the modular curves in the database listed below.
