Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ - x w + y z $ |
| $=$ | $z t + 4 z u + w u$ |
| $=$ | $x t + 4 x u + y u$ |
| $=$ | $2 x^{2} + 4 x y - 6 y^{2} - 4 z^{2} + 4 z w + t u + 2 u^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 400 x^{8} + 200 x^{6} y^{2} - 400 x^{6} z^{2} + 25 x^{4} y^{4} - 260 x^{4} y^{2} z^{2} + \cdots + 64 y^{4} z^{4} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -89x^{8} - 144x^{7} + 172x^{6} - 912x^{5} + 970x^{4} + 912x^{3} + 172x^{2} + 144x - 89 $ |
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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
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -\frac{2^2}{3^9}\cdot\frac{1600000zw^{9}-7680000zw^{7}u^{2}-46176000zw^{5}u^{4}-959289600zw^{3}u^{6}-5676236640zwu^{8}-1600000w^{10}+8480000w^{8}u^{2}-35936000w^{6}u^{4}-297635200w^{4}u^{6}-1801840800w^{2}u^{8}-19667t^{10}-196494t^{9}u-883127t^{8}u^{2}-2353320t^{7}u^{3}-4104598t^{6}u^{4}-3842564t^{5}u^{5}-95638t^{4}u^{6}+25295416t^{3}u^{7}+210459241t^{2}u^{8}+422897666tu^{9}+813155421u^{10}}{u^{6}(300zw^{3}-450zwu^{2}+100w^{4}-180w^{2}u^{2}-t^{4}-5t^{3}u-2t^{2}u^{2}+3tu^{3}+44u^{4})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
40.72.3.bn.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle w$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}u$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 400X^{8}+200X^{6}Y^{2}+25X^{4}Y^{4}-400X^{6}Z^{2}-260X^{4}Y^{2}Z^{2}-80X^{2}Y^{4}Z^{2}-10Y^{6}Z^{2}+676X^{4}Z^{4}+416X^{2}Y^{2}Z^{4}+64Y^{4}Z^{4} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
40.72.3.bn.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{200}{9}ywt^{2}-\frac{400}{3}ywtu-\frac{1000}{9}ywu^{2}+\frac{20}{9}yt^{3}+\frac{200}{9}yt^{2}u+\frac{580}{9}ytu^{2}+\frac{400}{9}yu^{3}-\frac{10}{3}wt^{3}-\frac{100}{3}wt^{2}u-\frac{290}{3}wtu^{2}-\frac{200}{3}wu^{3}+\frac{1}{3}t^{4}+\frac{14}{3}t^{3}u+23t^{2}u^{2}+\frac{136}{3}tu^{3}+\frac{80}{3}u^{4}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{496000}{243}ywt^{14}-\frac{178528000}{2187}ywt^{13}u-\frac{3178400000}{2187}ywt^{12}u^{2}-\frac{33140192000}{2187}ywt^{11}u^{3}-\frac{223275776000}{2187}ywt^{10}u^{4}-\frac{37185248000}{81}ywt^{9}u^{5}-\frac{2938850912000}{2187}ywt^{8}u^{6}-\frac{4696656800000}{2187}ywt^{7}u^{7}+\frac{131418608000}{243}ywt^{6}u^{8}+\frac{27568186496000}{2187}ywt^{5}u^{9}+\frac{75840129280000}{2187}ywt^{4}u^{10}+\frac{115607326720000}{2187}ywt^{3}u^{11}+\frac{105850572800000}{2187}ywt^{2}u^{12}+\frac{53694464000000}{2187}ywtu^{13}+\frac{1269760000000}{243}ywu^{14}+\frac{208000}{243}yt^{15}+\frac{74848000}{2187}yt^{14}u+\frac{1358176000}{2187}yt^{13}u^{2}+\frac{14764064000}{2187}yt^{12}u^{3}+\frac{106675072000}{2187}yt^{11}u^{4}+\frac{534931936000}{2187}yt^{10}u^{5}+\frac{207325088000}{243}yt^{9}u^{6}+\frac{1417711648000}{729}yt^{8}u^{7}+\frac{492282544000}{243}yt^{7}u^{8}-\frac{2752099904000}{729}yt^{6}u^{9}-\frac{47833710848000}{2187}yt^{5}u^{10}-\frac{111215713280000}{2187}yt^{4}u^{11}-\frac{159883816960000}{2187}yt^{3}u^{12}-\frac{146961612800000}{2187}yt^{2}u^{13}-\frac{79486976000000}{2187}ytu^{14}-\frac{2129920000000}{243}yu^{15}-\frac{8000}{81}wt^{15}-\frac{1360000}{243}wt^{14}u-\frac{863984000}{6561}wt^{13}u^{2}-\frac{11506640000}{6561}wt^{12}u^{3}-\frac{10854592000}{729}wt^{11}u^{4}-\frac{556244272000}{6561}wt^{10}u^{5}-\frac{2137191440000}{6561}wt^{9}u^{6}-\frac{5217832816000}{6561}wt^{8}u^{7}-\frac{5701523000000}{6561}wt^{7}u^{8}+\frac{10336314848000}{6561}wt^{6}u^{9}+\frac{59290384000000}{6561}wt^{5}u^{10}+\frac{43315417600000}{2187}wt^{4}u^{11}+\frac{167071078400000}{6561}wt^{3}u^{12}+\frac{126947532800000}{6561}wt^{2}u^{13}+\frac{1839104000000}{243}wtu^{14}+\frac{81920000000}{81}wu^{15}+\frac{4400}{81}t^{16}+\frac{666400}{243}t^{15}u+\frac{403220000}{6561}t^{14}u^{2}+\frac{593845600}{729}t^{13}u^{3}+\frac{192793600}{27}t^{12}u^{4}+\frac{286067194400}{6561}t^{11}u^{5}+\frac{1238242700000}{6561}t^{10}u^{6}+\frac{3723017140000}{6561}t^{9}u^{7}+\frac{6939674363600}{6561}t^{8}u^{8}+\frac{3126143603200}{6561}t^{7}u^{9}-\frac{26198139212800}{6561}t^{6}u^{10}-\frac{10795794227200}{729}t^{5}u^{11}-\frac{189819630592000}{6561}t^{4}u^{12}-\frac{236861997056000}{6561}t^{3}u^{13}-\frac{187120517120000}{6561}t^{2}u^{14}-\frac{3080192000000}{243}tu^{15}-\frac{180224000000}{81}u^{16}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -t^{4}-\frac{86}{9}t^{3}u-\frac{301}{9}t^{2}u^{2}-\frac{584}{9}tu^{3}-80u^{4}$ |
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.
