Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ 2 y^{2} - 2 y w - 3 t^{2} $ |
| $=$ | $2 x z + 2 x t + 4 x u + w t$ |
| $=$ | $ - y w + z^{2} - z t - 2 z u - 2 w^{2} - 2 t^{2} - 2 t u - 2 u^{2}$ |
| $=$ | $2 y^{2} + 2 y w - 3 z^{2} + 2 w^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{8} - 4 x^{6} y^{2} - 8 x^{6} y z - 28 x^{6} z^{2} + x^{4} y^{4} + 4 x^{4} y^{3} z + 6 x^{4} y^{2} z^{2} + \cdots + 36 z^{8} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 78x^{8} + 240x^{7} + 1248x^{6} + 1248x^{5} + 1680x^{4} - 2496x^{3} + 4992x^{2} - 1920x + 1248 $ |
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This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 3^3\,\frac{3745812xt^{8}+20540184xt^{7}u+46379664xt^{6}u^{2}+55821792xt^{5}u^{3}+40113600xt^{4}u^{4}+21566592xt^{3}u^{5}+5521152xt^{2}u^{6}+1577472xtu^{7}+49152yt^{8}+1248604yt^{7}u+4202092yt^{6}u^{2}+7038816yt^{5}u^{3}+6415312yt^{4}u^{4}+3660736yt^{3}u^{5}+1529664yt^{2}u^{6}+265216ytu^{7}+66304yu^{8}-536178zwt^{7}-1661532zwt^{6}u-1864392zwt^{5}u^{2}-1565424zwt^{4}u^{3}-482400zwt^{3}u^{4}-190272zwt^{2}u^{5}+2688zwtu^{6}+768zwu^{7}+1019605wt^{8}+3875544wt^{7}u+6197208wt^{6}u^{2}+4909696wt^{5}u^{3}+3121536wt^{4}u^{4}+802176wt^{3}u^{5}+270976wt^{2}u^{6}+3072wtu^{7}+768wu^{8}}{1740xt^{8}+20760xt^{7}u+102528xt^{6}u^{2}+249600xt^{5}u^{3}+314880xt^{4}u^{4}+211968xt^{3}u^{5}+86016xt^{2}u^{6}+24576xtu^{7}+580yt^{7}u+5380yt^{6}u^{2}+20352yt^{5}u^{3}+38080yt^{4}u^{4}+36352yt^{3}u^{5}+16896yt^{2}u^{6}+4096ytu^{7}+1024yu^{8}-414zwt^{7}-3132zwt^{6}u-8448zwt^{5}u^{2}-10752zwt^{4}u^{3}-7680zwt^{3}u^{4}-3072zwt^{2}u^{5}+451wt^{8}+5316wt^{7}u+20676wt^{6}u^{2}+34816wt^{5}u^{3}+27648wt^{4}u^{4}+12288wt^{3}u^{5}+4096wt^{2}u^{6}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
24.72.3.q.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 2u$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle t$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 4X^{8}-4X^{6}Y^{2}+X^{4}Y^{4}-8X^{6}YZ+4X^{4}Y^{3}Z-28X^{6}Z^{2}+6X^{4}Y^{2}Z^{2}+4X^{4}YZ^{3}+61X^{4}Z^{4}+6X^{2}Y^{2}Z^{4}+12X^{2}YZ^{5}-66X^{2}Z^{6}+36Z^{8} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
24.72.3.q.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle \frac{1}{2}y^{5}t-y^{5}u-\frac{17}{4}y^{3}t^{3}-\frac{3}{2}y^{3}t^{2}u+3y^{3}tu^{2}+2y^{3}u^{3}+6yt^{5}+6yt^{4}u$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 216y^{21}t^{3}-216y^{20}t^{4}-3348y^{19}t^{5}-1728y^{19}t^{4}u-432y^{19}t^{3}u^{2}+3672y^{18}t^{6}+2376y^{18}t^{5}u+432y^{18}t^{4}u^{2}+21573y^{17}t^{7}+18306y^{17}t^{6}u+6480y^{17}t^{5}u^{2}+864y^{17}t^{4}u^{3}-25110y^{16}t^{8}-25596y^{16}t^{7}u-7128y^{16}t^{6}u^{2}-1296y^{16}t^{5}u^{3}-\frac{155115}{2}y^{15}t^{9}-87561y^{15}t^{8}u-39690y^{15}t^{7}u^{2}-7452y^{15}t^{6}u^{3}+95094y^{14}t^{10}+124740y^{14}t^{9}u+46008y^{14}t^{8}u^{2}+11664y^{14}t^{7}u^{3}+\frac{338985}{2}y^{13}t^{11}+244215y^{13}t^{10}u+135594y^{13}t^{9}u^{2}+28188y^{13}t^{8}u^{3}-220644y^{12}t^{12}-355752y^{12}t^{11}u-163296y^{12}t^{10}u^{2}-46656y^{12}t^{9}u^{3}-220887y^{11}t^{13}-430110y^{11}t^{12}u-285768y^{11}t^{11}u^{2}-58320y^{11}t^{10}u^{3}+316386y^{10}t^{14}+644436y^{10}t^{13}u+355752y^{10}t^{12}u^{2}+104976y^{10}t^{11}u^{3}+\frac{269001}{2}y^{9}t^{15}+483327y^{9}t^{14}u+380538y^{9}t^{13}u^{2}+67068y^{9}t^{12}u^{3}-250776y^{8}t^{16}-752328y^{8}t^{15}u-489888y^{8}t^{14}u^{2}-139968y^{8}t^{13}u^{3}+47385y^{7}t^{17}-330966y^{7}t^{16}u-309096y^{7}t^{15}u^{2}-34992y^{7}t^{14}u^{3}+42282y^{6}t^{18}+545292y^{6}t^{17}u+414072y^{6}t^{16}u^{2}+104976y^{6}t^{15}u^{3}-\frac{298161}{2}y^{5}t^{19}+120285y^{5}t^{18}u+135594y^{5}t^{17}u^{2}-2916y^{5}t^{16}u^{3}+109350y^{4}t^{20}-218700y^{4}t^{19}u-192456y^{4}t^{18}u^{2}-34992y^{4}t^{17}u^{3}+\frac{207765}{2}y^{3}t^{21}-15309y^{3}t^{20}u-21870y^{3}t^{19}u^{2}+8748y^{3}t^{18}u^{3}-96228y^{2}t^{22}+34992y^{2}t^{21}u+34992y^{2}t^{20}u^{2}-26244yt^{23}+26244t^{24}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y^{5}t-\frac{3}{2}y^{4}t^{2}-3y^{3}t^{3}+\frac{9}{2}y^{2}t^{4}+3yt^{5}-\frac{9}{2}t^{6}$ |
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.
