Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ y^{2} - 2 y t - z w - w^{2} $ |
| $=$ | $2 y^{2} - z^{2} - w^{2}$ |
| $=$ | $3 x t + u^{2}$ |
| $=$ | $4 x y + y t + z w$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 162 x^{8} - 36 x^{6} y^{2} - 108 x^{6} z^{2} + x^{4} y^{4} + 36 x^{4} z^{4} - 4 x^{2} y^{2} z^{4} + \cdots + 2 z^{8} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{4} + 1\right) y $ | $=$ | $ -x^{8} + 21x^{6} - 53x^{4} + 21x^{2} - 1 $ |
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This modular curve has no $\Q_p$ points for $p=11$, and therefore no rational points.
Maps to other modular curves
$j$-invariant map
of degree 96 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{1}{2}\cdot\frac{23677920yt^{11}+112269888yt^{9}u^{2}+375881472yt^{7}u^{4}+748376064yt^{5}u^{6}+1053720576yt^{3}u^{8}+767311872ytu^{10}+51030zw^{11}+577368zw^{9}u^{2}+3893184zw^{7}u^{4}+16699392zw^{5}u^{6}+57305088zw^{3}u^{8}+202727424zwu^{10}+21141w^{12}+103032w^{10}u^{2}-51840w^{8}u^{4}-6594048w^{6}u^{6}-56687616w^{4}u^{8}-411033600w^{2}u^{10}-9593640t^{12}-158754816t^{10}u^{2}-433506816t^{8}u^{4}-1196937216t^{6}u^{6}-1678786560t^{4}u^{8}-1626144768t^{2}u^{10}-1668349952u^{12}}{u^{4}(38880yt^{7}+98496yt^{5}u^{2}+156672yt^{3}u^{4}+124416ytu^{6}+972zw^{7}+7344zw^{5}u^{2}+17856zw^{3}u^{4}+33024zwu^{6}+405w^{8}+432w^{6}u^{2}-14688w^{4}u^{4}-83712w^{2}u^{6}-54756t^{8}-138240t^{6}u^{2}-265536t^{4}u^{4}-245760t^{2}u^{6}-299008u^{8})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
48.96.3.jx.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 3w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2u$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 162X^{8}-36X^{6}Y^{2}+X^{4}Y^{4}-108X^{6}Z^{2}+36X^{4}Z^{4}-4X^{2}Y^{2}Z^{4}-12X^{2}Z^{6}+2Z^{8} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
48.96.3.jx.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -w^{3}t^{3}-w^{2}t^{4}+\frac{4}{3}w^{2}t^{2}u^{2}+2wt^{5}+\frac{8}{3}wt^{3}u^{2}+\frac{64}{9}wtu^{4}+t^{6}-\frac{4}{3}t^{4}u^{2}+\frac{16}{3}t^{2}u^{4}-\frac{64}{9}u^{6}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -30w^{3}t^{21}-96w^{3}t^{20}u+104w^{3}t^{19}u^{2}+256w^{3}t^{18}u^{3}+\frac{256}{9}w^{3}t^{17}u^{4}+\frac{1024}{9}w^{3}t^{16}u^{5}-\frac{10240}{27}w^{3}t^{15}u^{6}-\frac{4096}{3}w^{3}t^{14}u^{7}+\frac{11264}{27}w^{3}t^{13}u^{8}+\frac{32768}{27}w^{3}t^{12}u^{9}+\frac{4096}{81}w^{3}t^{11}u^{10}+\frac{65536}{27}w^{3}t^{10}u^{11}-\frac{262144}{243}w^{3}t^{9}u^{12}-\frac{262144}{81}w^{3}t^{8}u^{13}+\frac{262144}{243}w^{3}t^{7}u^{14}-\frac{1048576}{729}w^{3}t^{6}u^{15}+\frac{4849664}{6561}w^{3}t^{5}u^{16}+\frac{14680064}{6561}w^{3}t^{4}u^{17}-\frac{19398656}{19683}w^{3}t^{3}u^{18}-56w^{2}t^{22}-176w^{2}t^{21}u+\frac{512}{3}w^{2}t^{20}u^{2}+448w^{2}t^{19}u^{3}+\frac{128}{3}w^{2}t^{18}u^{4}+\frac{1024}{3}w^{2}t^{17}u^{5}-\frac{16384}{27}w^{2}t^{16}u^{6}-\frac{53248}{27}w^{2}t^{15}u^{7}+\frac{53248}{81}w^{2}t^{14}u^{8}+\frac{40960}{27}w^{2}t^{13}u^{9}+\frac{163840}{81}w^{2}t^{11}u^{11}-\frac{851968}{729}w^{2}t^{10}u^{12}-\frac{3407872}{729}w^{2}t^{9}u^{13}+\frac{4194304}{2187}w^{2}t^{8}u^{14}+\frac{1048576}{729}w^{2}t^{7}u^{15}-\frac{524288}{2187}w^{2}t^{6}u^{16}+\frac{7340032}{2187}w^{2}t^{5}u^{17}-\frac{33554432}{19683}w^{2}t^{4}u^{18}-\frac{46137344}{19683}w^{2}t^{3}u^{19}+\frac{58720256}{59049}w^{2}t^{2}u^{20}+16wt^{23}+64wt^{22}u-\frac{400}{3}wt^{21}u^{2}-256wt^{20}u^{3}+\frac{1664}{9}wt^{19}u^{4}+\frac{8192}{9}wt^{18}u^{5}-\frac{3328}{27}wt^{17}u^{6}+\frac{16384}{27}wt^{16}u^{7}-\frac{65536}{81}wt^{15}u^{8}-\frac{32768}{9}wt^{14}u^{9}+\frac{57344}{27}wt^{13}u^{10}+\frac{131072}{81}wt^{12}u^{11}-\frac{65536}{729}wt^{11}u^{12}-\frac{1048576}{729}wt^{10}u^{13}-\frac{3276800}{2187}wt^{9}u^{14}-\frac{4194304}{729}wt^{8}u^{15}+\frac{19922944}{6561}wt^{7}u^{16}+\frac{104857600}{6561}wt^{6}u^{17}-\frac{105906176}{19683}wt^{5}u^{18}+\frac{83886080}{19683}wt^{4}u^{19}-\frac{159383552}{59049}wt^{3}u^{20}-\frac{268435456}{19683}wt^{2}u^{21}+\frac{117440512}{19683}wtu^{22}+31t^{24}+112t^{23}u-\frac{656}{3}t^{22}u^{2}-\frac{1600}{3}t^{21}u^{3}+\frac{1184}{3}t^{20}u^{4}+\frac{10496}{9}t^{19}u^{5}-\frac{1280}{27}t^{18}u^{6}+\frac{1024}{3}t^{17}u^{7}-\frac{93952}{81}t^{16}u^{8}-\frac{434176}{81}t^{15}u^{9}+\frac{237568}{81}t^{14}u^{10}+\frac{1409024}{243}t^{13}u^{11}-\frac{475136}{243}t^{12}u^{12}+\frac{131072}{729}t^{11}u^{13}-\frac{5373952}{2187}t^{10}u^{14}-\frac{16252928}{2187}t^{9}u^{15}+\frac{12255232}{2187}t^{8}u^{16}+\frac{120586240}{6561}t^{7}u^{17}-\frac{139460608}{19683}t^{6}u^{18}-\frac{20971520}{2187}t^{5}u^{19}+\frac{98566144}{59049}t^{4}u^{20}-\frac{989855744}{59049}t^{3}u^{21}+\frac{486539264}{59049}t^{2}u^{22}+\frac{2483027968}{177147}tu^{23}-\frac{3238002688}{531441}u^{24}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -t^{6}-\frac{4}{3}t^{4}u^{2}+\frac{16}{9}t^{2}u^{4}+\frac{64}{27}u^{6}$ |
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.
