Embedded model Embedded model in $\mathbb{P}^{6}$
| $ 0 $ | $=$ | $ y^{2} v - y u v - z^{2} v + z t v + w t v - u^{2} v $ |
| $=$ | $x y v - x t v + x u v - y^{2} v - z w v - z t v + z u v - w t v + w u v$ |
| $=$ | $x w v - x t v - y t v + z t v + w^{2} v + w t v - w u v - t u v$ |
| $=$ | $ - y w v + y t v + z^{2} v - z u v - w^{2} v - w t v + w u v + t^{2} v$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 11 x^{7} - 24 x^{6} z + 1845 x^{5} y^{2} - 6 x^{5} z^{2} - 5700 x^{4} y^{2} z + 45 x^{4} z^{3} + \cdots + 2 z^{7} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -3x^{11} + 33x^{6} + 3x $ |
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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}{2^9\cdot5\cdot401^9}\cdot\frac{701222862741385621719403211454432xtu^{9}+430381781962133226092476362437256xtu^{7}v^{2}-4231576215490406588265272351916600xtu^{5}v^{4}+228754935730839555441226416619584xtu^{3}v^{6}-72670611390343427976016290010752xtuv^{8}-236867099642816718545014873852656xu^{10}+633713318038346003588804404320072xu^{8}v^{2}+1539060739061073829758777044420100xu^{6}v^{4}-686981915710170575081163901137792xu^{4}v^{6}+57683329022236137326415229277376xu^{2}v^{8}-8721322022796028668704364242560xv^{10}-596617811437760860861367058267441ytu^{9}-1435085821845027189689853973375458ytu^{7}v^{2}+3938361836566633971314599611165240ytu^{5}v^{4}-303866360388917810374166771850432ytu^{3}v^{6}+48639890500812701786959462457856ytuv^{8}+1559232046956536009388598008337920yu^{10}+1098808997390088704400152395410525yu^{8}v^{2}+1275962717437187805246630653684100yu^{6}v^{4}+109047486375521880121952859308400yu^{4}v^{6}-13826186856820446071825991640800yu^{2}v^{8}+7905128724951221146667525156480yv^{10}+520768019976676964156358791791947ztu^{9}-2116336658073219930953084409747144ztu^{7}v^{2}-9529338717226558397432259369964440ztu^{5}v^{4}+422840863505721176472889098563904ztu^{3}v^{6}-126505707218289195615012672348672ztuv^{8}-574676775891681082177329167286843zu^{10}+1107877719836670400055611569588426zu^{8}v^{2}+5941799718965484404282346592692900zu^{6}v^{4}-1823557489528761297836249930793936zu^{4}v^{6}+124245616323254136499189347076608zu^{2}v^{8}+6506934735249858560384427783040zv^{10}+1992312073691505243029086902966843wtu^{9}+824220214294850311777732666633014wtu^{7}v^{2}-10439932215854612684560198680963360wtu^{5}v^{4}+478764230870548262185599405751776wtu^{3}v^{6}-122763372675648949434356388403968wtuv^{8}-489239892935638741054513716250527wu^{10}+3398721500015378410438418015956689wu^{8}v^{2}+2607992195208618130191227940971400wu^{6}v^{4}-1163726695185622951466448014675904wu^{4}v^{6}+134128875439523850425062361265312wu^{2}v^{8}+1957009676791621122581744629760wv^{10}+356455395770714607168004429916316t^{2}u^{9}+129258479766284774181620653280868t^{2}u^{7}v^{2}-5756682711740615624087311494287520t^{2}u^{5}v^{4}+441172616645055508416735876057312t^{2}u^{3}v^{6}-72300635232288008125000198050816t^{2}uv^{8}-1283494424791593162603208035126843tu^{10}-1897342321134244573784208621047094tu^{8}v^{2}+6099484289021771650977108289416480tu^{6}v^{4}-1287536165403351484406783811019056tu^{4}v^{6}+62526996518575497495391986161088tu^{2}v^{8}+406480730147169508941425477120tv^{10}+1200525540900156012827539132294896u^{11}-353524478528934376742778192805797u^{9}v^{2}-2016697815909105033522815544761100u^{7}v^{4}+431104302293213519873013609254592u^{5}v^{6}-67948525521901186220452288927776u^{3}v^{8}+217399260522118496024138555520uv^{10}}{v^{10}(y+14z+11w+7t-3u)}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
30.120.5.e.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{3}v$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 11X^{7}+1845X^{5}Y^{2}-24X^{6}Z-5700X^{4}Y^{2}Z-6X^{5}Z^{2}+7050X^{3}Y^{2}Z^{2}+45X^{4}Z^{3}-4350X^{2}Y^{2}Z^{3}-65X^{3}Z^{4}+1350XY^{2}Z^{4}+33X^{2}Z^{5}-165Y^{2}Z^{5}-7XZ^{6}+2Z^{7} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
30.120.5.e.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{3}{5}x+\frac{1}{5}y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{123}{625}x^{5}v+\frac{76}{125}x^{4}yv-\frac{94}{125}x^{3}y^{2}v+\frac{58}{125}x^{2}y^{3}v-\frac{18}{125}xy^{4}v+\frac{11}{625}y^{5}v$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{1}{5}x+\frac{2}{5}y$ |
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.
