Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} t - x y t + x z t + y^{2} t $ |
| $=$ | $2 x y t + x w t + y z t$ |
| $=$ | $x^{2} w - x y w + x z w + y^{2} w$ |
| $=$ | $2 x y w + x w^{2} + y z w$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 14 x^{7} - 29 x^{6} z + x^{5} y^{2} + 21 x^{5} z^{2} + 7 x^{4} z^{3} - 28 x^{3} z^{4} + 21 x^{2} z^{5} + \cdots + z^{7} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -x^{7} - 7x^{4} + 8x $ |
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This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:0:0:0:1)$, $(-1:1:3:1:0)$, $(-1:-1:1:1:0)$, $(1/2:1:-3/2:1:0)$ |
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{1}{2^4\cdot7^2\cdot31^2}\cdot\frac{303859816772302189513355xzw^{9}+122598921434806393805328xzw^{7}t^{2}+30108627658371555589320xzw^{5}t^{4}+15985825178271498608352xzw^{3}t^{6}+1177977050157526525088xzwt^{8}+25396318339408801953xw^{10}-28795468085361931956224xw^{8}t^{2}-8988828419497746386176xw^{6}t^{4}-905032672186460380232xw^{4}t^{6}-1110128663765980962544xw^{2}t^{8}-75959015909049680960xt^{10}-105091643137464842266362yzw^{9}-186918936813701057856368yzw^{7}t^{2}-74244255962035436410752yzw^{5}t^{4}-12697103831720729281392yzw^{3}t^{6}-819566337198924710864yzwt^{8}+24742017040262046288944yw^{10}+34067904856573945653052yw^{8}t^{2}+22907314974217480858616yw^{6}t^{4}+8812774667196618815256yw^{4}t^{6}+2334976543789190911792yw^{2}t^{8}+141808722595283093280yt^{10}+43728781243088464013544z^{2}w^{9}-106241324952466836522448z^{2}w^{7}t^{2}-52051652283493404516292z^{2}w^{5}t^{4}-4914429816355697934016z^{2}w^{3}t^{6}-409168639585357649872z^{2}wt^{8}+43475385665842096957615zw^{10}-30093063543619295622305zw^{8}t^{2}-13078219375158926623770zw^{6}t^{4}+856506450736323654828zw^{4}t^{6}-93519765546417545440zw^{2}t^{8}-77431635020899930960zt^{10}+12412518650045904129739w^{11}+525478427734881440278w^{9}t^{2}-888188569922938872540w^{7}t^{4}+398707629157277671504w^{5}t^{6}+92446976545427696976w^{3}t^{8}-65926930198547192320wt^{10}}{t^{2}(6313665165312xzw^{7}+13270250658213xzw^{5}t^{2}-2862480868960xzw^{3}t^{4}+96892293288xzwt^{6}-2186561503249xw^{6}t^{2}-5128043752312xw^{4}t^{4}+350995028464xw^{2}t^{6}+2872085272xt^{8}+59973353828352yzw^{7}+139619789606490yzw^{5}t^{2}-10199838111576yzw^{3}t^{4}+222740465920yzwt^{6}+6313147633664yw^{8}+12884918004432yw^{6}t^{2}-5873736475428yw^{4}t^{4}+203343816592yw^{2}t^{6}-11084264520yt^{8}+50504252635136z^{2}w^{7}+116723251034520z^{2}w^{5}t^{2}-9669925219032z^{2}w^{3}t^{4}+186829187132z^{2}wt^{6}+18939340175360zw^{8}+41083132077505zw^{6}t^{2}-10144623393159zw^{4}t^{4}+504289482058zw^{2}t^{6}-630502180zt^{8}+3156573816832w^{9}+7360223030117w^{7}t^{2}-490052706638w^{5}t^{4}+630502180w^{3}t^{6})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
36.72.3.w.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 14X^{7}+X^{5}Y^{2}-29X^{6}Z+21X^{5}Z^{2}+7X^{4}Z^{3}-28X^{3}Z^{4}+21X^{2}Z^{5}-7XZ^{6}+Z^{7} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
36.72.3.w.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -x+y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle x^{3}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle x$ |
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.
