Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x z w + y w^{2} $ |
| $=$ | $x z^{2} + y z w$ |
| $=$ | $x y z + y^{2} w$ |
| $=$ | $x^{2} z + x y w$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{5} + x^{4} z + 3 x^{3} y^{2} + x^{3} z^{2} - 9 x^{2} y^{2} z + x^{2} z^{3} + 3 x y^{2} z^{2} + 3 y^{2} z^{3} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{3} + x^{2} + x + 1\right) y $ | $=$ | $ -x^{6} + x^{5} - x^{3} - 2x - 1 $ |
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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{31154869771466760x^{2}y^{18}+965342835625366512x^{2}y^{16}w^{2}+6940892689176456216x^{2}y^{14}w^{4}-2306584347902584704x^{2}y^{12}w^{6}+4131576573985940976x^{2}y^{10}w^{8}-17512281643223045088x^{2}y^{8}w^{10}+86016408937201591056x^{2}y^{6}w^{12}-367744959503392975488x^{2}y^{4}w^{14}+666616975716533524488x^{2}y^{2}w^{16}-875354797702127992080x^{2}w^{18}+44065282871726064xy^{19}+1405024388002793520xy^{17}w^{2}+10770551279872697088xy^{15}w^{4}+122269672867794624xy^{13}w^{6}-5120762085188933472xy^{11}w^{8}+22997670231128755680xy^{9}w^{10}-113546510976255466752xy^{7}w^{12}+479872646339843034432xy^{5}w^{14}-779372811052115210640xy^{3}w^{16}+1067517111624548273904xyw^{18}+12910479999469560y^{20}+451844095999480128y^{18}w^{2}+4186530139063328856y^{16}w^{4}+4675386714646382784y^{14}w^{6}-11575612313564506416y^{12}w^{8}+47370973188310287552y^{10}w^{10}-230745283500163287600y^{8}w^{12}+1002163362346835871552y^{6}w^{14}-2078081901033112665288y^{4}w^{16}+2665175271363192843264y^{2}w^{18}-218640077003z^{20}-7999353970443z^{19}w-131379226454496z^{18}w^{2}-1332864639594743z^{17}w^{3}-9683190237681151z^{16}w^{4}-54636733952842480z^{15}w^{5}-250955484262444352z^{14}w^{6}-964803326436548056z^{13}w^{7}-3150009572150174422z^{12}w^{8}-8773196914677377086z^{11}w^{9}-20701565767833944408z^{10}w^{10}-40624551050934828542z^{9}w^{11}-63452989430999876998z^{8}w^{12}-73662737062999474568z^{7}w^{13}-51534797640950484584z^{6}w^{14}-27874251827909640864z^{5}w^{15}-66906361300622558719z^{4}w^{16}-342813397569040662591z^{3}w^{17}-352816526005827846216z^{2}w^{18}-355839037208182789267zw^{19}-291784932567375991299w^{20}}{w^{2}(3716622792x^{2}y^{14}w^{2}+8461170576x^{2}y^{12}w^{4}-28027956312x^{2}y^{10}w^{6}+403924436640x^{2}y^{8}w^{8}-4991917554504x^{2}y^{6}w^{10}+80820612924048x^{2}y^{4}w^{12}-1405697711647464x^{2}y^{2}w^{14}+25272641210210304x^{2}w^{16}-7433245584xy^{15}w^{2}-9489095568xy^{13}w^{4}+44071187616xy^{11}w^{6}-546213479328xy^{9}w^{8}+6856679646000xy^{7}w^{10}-111581501050512xy^{5}w^{12}+1943280009978624xy^{3}w^{14}-34960172583409152xyw^{16}-3716622792y^{16}w^{2}-25805410272y^{14}w^{4}+58745368584y^{12}w^{6}-1040809061952y^{10}w^{8}+12621913820712y^{8}w^{10}-202662860085216y^{6}w^{12}+3517463137161240y^{4}w^{14}-63175668127352064y^{2}w^{16}+35437z^{18}+1092421z^{17}w+17008706z^{16}w^{2}+179587659z^{15}w^{3}+1459953864z^{14}w^{4}+9866552873z^{13}w^{5}+58321915494z^{12}w^{6}+311983173927z^{11}w^{7}+1540067116990z^{10}w^{8}+7092458182111z^{9}w^{9}+30537694920238z^{8}w^{10}+123174478154257z^{7}w^{11}+460086073355512z^{6}w^{12}+1595621952627955z^{5}w^{13}+4801305369010538z^{4}w^{14}+13132628498675901z^{3}w^{15}+12794487539129941z^{2}w^{16}+11653390861136384zw^{17}+8424213736736768w^{18})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
48.96.2.h.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{5}+3X^{3}Y^{2}+X^{4}Z-9X^{2}Y^{2}Z+X^{3}Z^{2}+3XY^{2}Z^{2}+X^{2}Z^{3}+3Y^{2}Z^{3} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
48.96.2.h.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{3}{2}yz^{5}+\frac{9}{2}yz^{4}w-\frac{3}{2}yz^{3}w^{2}-\frac{3}{2}yz^{2}w^{3}-\frac{1}{2}z^{6}-\frac{1}{2}z^{5}w-\frac{1}{2}z^{4}w^{2}-\frac{1}{2}z^{3}w^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle zw$ |
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.
