Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ 2 x y w - y w t - z w t $ |
| $=$ | $2 x y t - y t^{2} - z t^{2}$ |
| $=$ | $2 x y^{2} - y^{2} t - y z t$ |
| $=$ | $2 x y z - y z t - z^{2} t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 3 x^{5} + 3 x^{4} z + 6 x^{3} y^{2} + 4 x^{3} z^{2} - 6 x^{2} y^{2} z + 4 x^{2} z^{3} - 2 x y^{2} z^{2} + \cdots + z^{5} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -2x^{7} - 10x^{6} - 14x^{5} - 20x^{4} - 14x^{3} - 10x^{2} - 2x $ |
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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 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{2}\cdot\frac{4518019316716380344xzt^{12}-3134504646475776xt^{13}-29896998912yz^{13}-47325250584576yz^{12}t+870541196525568yz^{11}t^{2}+193533217066450944yz^{10}t^{3}+165092842793631744yz^{9}t^{4}-2688146105723338752yz^{8}t^{5}-4347628399859134464yz^{7}t^{6}-2139002645660098560yz^{6}t^{7}+946285669184173056yz^{5}t^{8}+410616806574315328yz^{4}t^{9}-2905688277021581200yz^{3}t^{10}-9810497743927686680yz^{2}t^{11}-3908720293341134332yzt^{12}-578587880338774332yt^{13}+88332042240z^{14}-5122918711296z^{13}t-2866069558001664z^{12}t^{2}+14873450217209856z^{11}t^{3}+683718654253105152z^{10}t^{4}+1155997015680614400z^{9}t^{5}+2051020158744944640z^{8}t^{6}+2945123999293083648z^{7}t^{7}+964592686465786368z^{6}t^{8}-1891302262838945856z^{5}t^{9}-3122442562288901264z^{4}t^{10}+1038781170261809472z^{3}t^{11}-1097845667641152z^{2}w^{12}+14760667961167104z^{2}w^{10}t^{2}-63342995623560576z^{2}w^{8}t^{4}+1185219548985275952z^{2}w^{6}t^{6}-159218702889773760z^{2}w^{4}t^{8}-1718883831785641028z^{2}w^{2}t^{10}+4487456400558187824z^{2}t^{12}-3573842188388544zw^{12}t-7956414716945472zw^{10}t^{3}-230547368553587712zw^{8}t^{5}-342665576238649200zw^{6}t^{7}+1344071255264669184zw^{4}t^{9}+4706753351433466664zw^{2}t^{11}+842471615812254608zt^{13}+136372012128w^{14}+962489666406960w^{12}t^{2}+551716851159072w^{10}t^{4}+60007267799460648w^{8}t^{6}-20463717281328180w^{6}t^{8}-388803906194438622w^{4}t^{10}-765822165640356603w^{2}t^{12}-1155130346962944t^{14}}{t(1999110393090616xzt^{11}-33124515840yz^{12}-161418313728yz^{11}t+2672718446592yz^{10}t^{2}+9282767339520yz^{9}t^{3}-22399269396480yz^{8}t^{4}-108373016641536yz^{7}t^{5}-9852898975488yz^{6}t^{6}+524071640439360yz^{5}t^{7}+749862945045216yz^{4}t^{8}-1059074639385712yz^{3}t^{9}-4437091221003632yz^{2}t^{10}-1649306172349816yzt^{11}-258164801813156yt^{12}-3737124864z^{13}+270347010048z^{12}t+1005153878016z^{11}t^{2}-6854804029440z^{10}t^{3}-25518380507136z^{9}t^{4}+26482693588992z^{8}t^{5}+197302992324864z^{7}t^{6}+105841533846720z^{6}t^{7}-794923101266112z^{5}t^{8}-1745015868603168z^{4}t^{9}+403154338580432z^{3}t^{10}+8236007424z^{2}w^{10}t+59841607680z^{2}w^{8}t^{3}+51629033404044z^{2}w^{6}t^{5}+37747151773488z^{2}w^{4}t^{7}-858209868998556z^{2}w^{2}t^{9}+2096732396485552z^{2}t^{11}+58392576zw^{12}+78616811520zw^{10}t^{2}+4672004023296zw^{8}t^{4}+109401912654996zw^{6}t^{6}+799062568408404zw^{4}t^{8}+2187817600279952zw^{2}t^{10}+370695197366076zt^{12}-14598144w^{12}t-20161654272w^{10}t^{3}-1152854106426w^{8}t^{5}-29932809560517w^{6}t^{7}-183476303877432w^{4}t^{9}-342562598477846w^{2}t^{11})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
24.96.3.gm.3
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 3X^{5}+6X^{3}Y^{2}+3X^{4}Z-6X^{2}Y^{2}Z+4X^{3}Z^{2}-2XY^{2}Z^{2}+4X^{2}Z^{3}+2Y^{2}Z^{3}+XZ^{4}+Z^{5} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
24.96.3.gm.3
:
| $\displaystyle X$ |
$=$ |
$\displaystyle \frac{1}{2}y+\frac{1}{2}z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{3}{8}y^{3}w-\frac{3}{8}y^{2}zw-\frac{1}{8}yz^{2}w+\frac{1}{8}z^{3}w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}y-\frac{1}{2}z$ |
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.
