Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ 2 x^{2} t + 2 x y t - 3 x z t + x w t + y w t - z w t $ |
| $=$ | $x^{2} t + x z t - 2 x w t + 2 y^{2} t - 2 y z t - y w t + z w t$ |
| $=$ | $x^{3} - 2 x^{2} y + x^{2} z + x^{2} w + x y z - x z w - x w^{2} + y^{2} w - y z w$ |
| $=$ | $x^{3} + x^{2} y + x y z - x z^{2} - x z w - x w^{2} + 2 x t^{2} + y^{2} z - y z^{2} - y w^{2} + \cdots + w t^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 6 x^{5} z^{2} - 324 x^{4} y^{2} z - 4 x^{4} z^{3} + 4860 x^{3} y^{4} + 612 x^{3} y^{2} z^{2} + \cdots - 3 z^{7} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -3x^{7} + 15x^{6} - 21x^{5} + 30x^{4} - 21x^{3} + 15x^{2} - 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 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{2^{10}\cdot3}\cdot\frac{23589317363328xzt^{12}-69425394500448xw^{13}-391539768435519xw^{11}t^{2}-79749147528222xw^{9}t^{4}-117592132343652xw^{7}t^{6}+64917044662392xw^{5}t^{8}+36708599699280xw^{3}t^{10}-188910357544512xwt^{12}-306110016yz^{13}+3877393536yz^{11}t^{2}-12108351744yz^{9}t^{4}-47163617280yz^{7}t^{6}-136774490112yz^{5}t^{8}+9148169631744yz^{3}t^{10}-63163923990912yzt^{12}-3966981734016yw^{13}-72982734059202yw^{11}t^{2}-48265226082720yw^{9}t^{4}-30461487841368yw^{7}t^{6}+65165654958912yw^{5}t^{8}+233709331132512yw^{3}t^{10}+158058520611840ywt^{12}+306110016z^{14}-3061100160z^{12}t^{2}+136048896z^{10}t^{4}+132058128384z^{8}t^{6}-305837918208z^{6}t^{8}-5058237487104z^{4}t^{10}+86484752637120z^{2}w^{12}+3957728289468z^{2}w^{10}t^{2}+32556062388960z^{2}w^{8}t^{4}+27299065444272z^{2}w^{6}t^{6}+39039244154496z^{2}w^{4}t^{8}-15499162624320z^{2}w^{2}t^{10}+1728953460288z^{2}t^{12}+95212153266624zw^{13}+88047138140352zw^{11}t^{2}+80963537165280zw^{9}t^{4}+1408764286080zw^{7}t^{6}-103458499027776zw^{5}t^{8}-148428909582336zw^{3}t^{10}+27893217049920zwt^{12}+21819470922144w^{14}-84239546744001w^{12}t^{2}+48588004436906w^{10}t^{4}+12110700202596w^{8}t^{6}+5802945351192w^{6}t^{8}-66959835744528w^{4}t^{10}-89400832605504w^{2}t^{12}-4478976t^{14}}{t^{4}(20318688xzt^{8}-3xw^{9}+1594239xw^{7}t^{2}+44860131xw^{5}t^{4}+19496997xw^{3}t^{6}-85217832xwt^{8}-314928yz^{5}t^{4}+5668704yz^{3}t^{6}-28530144yzt^{8}-12yw^{9}-3189348yw^{7}t^{2}+27311256yw^{5}t^{4}+98076582yw^{3}t^{6}+60777216ywt^{8}+314928z^{6}t^{4}-4828896z^{4}t^{6}-6z^{2}w^{8}+3188286z^{2}w^{6}t^{2}+13812714z^{2}w^{4}t^{4}-10667538z^{2}w^{2}t^{6}+1574640z^{2}t^{8}+18zw^{9}+1026zw^{7}t^{2}-41125806zw^{5}t^{4}-57480192zw^{3}t^{6}+15664752zwt^{8}+3w^{10}-1594097w^{8}t^{2}-5841147w^{6}t^{4}-24499521w^{4}t^{6}-35408610w^{2}t^{8})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
24.96.3.gs.3
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{3}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 4860X^{3}Y^{4}-324X^{4}Y^{2}Z-11340X^{2}Y^{4}Z+6X^{5}Z^{2}+612X^{3}Y^{2}Z^{2}+5940XY^{4}Z^{2}-4X^{4}Z^{3}-420X^{2}Y^{2}Z^{3}-900Y^{4}Z^{3}+13X^{3}Z^{4}+264XY^{2}Z^{4}-13X^{2}Z^{5}+180Y^{2}Z^{5}-15XZ^{6}-3Z^{7} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
24.96.3.gs.3
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{1}{15}y^{4}w+\frac{4}{5}y^{3}w^{2}+\frac{2}{3}y^{3}t^{2}-\frac{59}{135}y^{2}w^{3}-\frac{14}{9}y^{2}wt^{2}-\frac{32}{135}yw^{4}+\frac{22}{27}yw^{2}t^{2}-\frac{10}{81}w^{3}t^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{644}{50625}y^{15}w^{4}t+\frac{14176}{151875}y^{14}w^{5}t-\frac{328}{10125}y^{14}w^{3}t^{3}+\frac{12772}{151875}y^{13}w^{6}t-\frac{6128}{30375}y^{13}w^{4}t^{3}-\frac{840884}{1366875}y^{12}w^{7}t+\frac{3472}{91125}y^{12}w^{5}t^{3}-\frac{3025924}{4100625}y^{11}w^{8}t+\frac{53048}{30375}y^{11}w^{6}t^{3}+\frac{2347732}{1366875}y^{10}w^{9}t+\frac{201992}{820125}y^{10}w^{7}t^{3}+\frac{15060328}{12301875}y^{9}w^{10}t-\frac{14255872}{2460375}y^{9}w^{8}t^{3}-\frac{7815308}{4100625}y^{8}w^{11}t+\frac{9497128}{7381125}y^{8}w^{9}t^{3}-\frac{2630036}{4100625}y^{7}w^{12}t+\frac{45624376}{7381125}y^{7}w^{10}t^{3}+\frac{8984104}{12301875}y^{6}w^{13}t-\frac{13502048}{7381125}y^{6}w^{11}t^{3}+\frac{95092}{4100625}y^{5}w^{14}t-\frac{19546696}{7381125}y^{5}w^{12}t^{3}-\frac{103036}{820125}y^{4}w^{15}t+\frac{199448}{295245}y^{4}w^{13}t^{3}+\frac{2836}{32805}y^{3}w^{16}t+\frac{135296}{295245}y^{3}w^{14}t^{3}+\frac{1804}{32805}y^{2}w^{17}t-\frac{6464}{59049}y^{2}w^{15}t^{3}+\frac{8}{2187}yw^{18}t-\frac{1640}{59049}yw^{16}t^{3}-\frac{20}{19683}w^{19}t+\frac{400}{59049}w^{17}t^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{7}{15}y^{4}w+\frac{29}{45}y^{3}w^{2}+\frac{4}{3}y^{3}t^{2}-\frac{22}{135}y^{2}w^{3}-\frac{28}{9}y^{2}wt^{2}+\frac{7}{27}yw^{4}+\frac{44}{27}yw^{2}t^{2}+\frac{1}{9}w^{5}-\frac{20}{81}w^{3}t^{2}$ |
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.
