Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x y^{2} + 2 x y z + x y w + y^{3} - y^{2} z + y^{2} w - y z^{2} + y z w + 2 z^{3} + z^{2} w $ |
| $=$ | $x y^{2} - x y w - y^{3} - 2 y^{2} w + 4 y z^{2} - y w^{2}$ |
| $=$ | $x^{2} y - x y z + x y w + x z^{2} + y^{3} - 2 y^{2} z + 2 y^{2} w - 2 y z w + y w^{2}$ |
| $=$ | $x y^{2} + x y z + x y w + x z w + y^{3} + y^{2} w - y z^{2} + 3 y z w - 2 z^{3} + z^{2} w + z w^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 2 x^{4} + x^{3} y + 3 x^{3} z - 5 x^{2} y^{2} + 6 x^{2} y z + 2 x^{2} z^{2} - 6 x y^{3} + \cdots - y^{2} z^{2} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{2} + x\right) y $ | $=$ | $ x^{6} - 3x^{5} + 6x^{4} - 8x^{3} + 6x^{2} - 3x + 1 $ |
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This modular curve has 6 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Embedded model |
|---|
| $(0:-1:0:1)$, $(1/2:-1/2:1/2:1)$, $(1:0:0:0)$, $(-1:0:0:1)$, $(-1:1:1:1)$, $(0:0:-1/2:1)$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle -3\,\frac{5008426795008x^{10}-22634236477440x^{9}z+24367922675712x^{9}w-53359008546816x^{8}zw+38911623561216x^{8}w^{2}-58560067141632x^{7}zw^{2}+32790213033984x^{7}w^{3}-40837941559296x^{6}zw^{3}+22706770673664x^{6}w^{4}-2358274507416576x^{5}zw^{4}-1209548649738240x^{5}w^{5}+22306833811196928x^{4}zw^{5}+18192961603553280x^{4}w^{6}+38398604279166752x^{3}zw^{6}-15623411137975712x^{3}w^{7}+2432355423805892x^{2}zw^{7}-32829140341326517x^{2}w^{8}+189599554173625344xz^{9}+66176393521932288xz^{8}w-1028333486184238080xz^{7}w^{2}-545127327389260800xz^{6}w^{3}+365145024994737120xz^{5}w^{4}+626910665797517984xz^{4}w^{5}-84785638599898540xz^{3}w^{6}-218042342780958445xz^{2}w^{7}+109419570029704523xzw^{8}-37143481555477727xw^{9}-33791680096059919y^{2}w^{8}+126721196224939008yz^{9}-441050065082855424yz^{8}w-886093154669786112yz^{7}w^{2}+322409985823448064yz^{6}w^{3}+680811059998208736yz^{5}w^{4}+304996813852998240yz^{4}w^{5}-482141075002789228yz^{3}w^{6}+46282289454473575yz^{2}w^{7}+108594543334801201yzw^{8}-73103511080186329yw^{9}-155686164491999232z^{10}+1051106553239049216z^{9}w+683703187255388160z^{8}w^{2}-1750614116203330560z^{7}w^{3}-1364596939347119424z^{6}w^{4}+648708673990469920z^{5}w^{5}+1163885470942954856z^{4}w^{6}-530900017712072486z^{3}w^{7}-47709446343759039z^{2}w^{8}+120745945332226727zw^{9}-39353439452884938w^{10}}{4081839381504x^{5}zw^{4}+4886575423488x^{5}w^{5}-30645958652928x^{4}zw^{5}-21512576808960x^{4}w^{6}-50964446096096x^{3}zw^{6}+5768109449696x^{3}w^{7}-7204163652092x^{2}zw^{7}+7122489037099x^{2}w^{8}-330628989901824xz^{9}-333960830724096xz^{8}w+1578651649348608xz^{7}w^{2}+1000299314414592xz^{6}w^{3}-621452206617120xz^{5}w^{4}-608605481337056xz^{4}w^{5}+183983104586644xz^{3}w^{6}+306223780092979xz^{2}w^{7}-93435884675861xzw^{8}-33349371489343xw^{9}-29995420060463y^{2}w^{8}-216503839374336yz^{9}+602545605083136yz^{8}w+1611440487336960yz^{7}w^{2}-190331052106752yz^{6}w^{3}-1050521700985632yz^{5}w^{4}-40870585110816yz^{4}w^{5}+576464923458772yz^{3}w^{6}+26175502310855yz^{2}w^{7}-59086726653103yzw^{8}-38300018904761yw^{9}+255283317467136z^{10}-1557013511752704z^{9}w-1846619142432768z^{8}w^{2}+2770407011386368z^{7}w^{3}+1925247215888064z^{6}w^{4}-1229556901877728z^{5}w^{5}-1209462663370136z^{4}w^{6}+610877203701722z^{3}w^{7}+264330635858913z^{2}w^{8}-102468369085433zw^{9}-8304598844298w^{10}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
14.48.2.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 2X^{4}+X^{3}Y-5X^{2}Y^{2}-6XY^{3}+3X^{3}Z+6X^{2}YZ-2XY^{2}Z-2Y^{3}Z+2X^{2}Z^{2}+XYZ^{2}-Y^{2}Z^{2}+XZ^{3} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
14.48.2.a.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle \frac{1}{2}y^{2}-\frac{1}{3}yw-\frac{1}{6}w^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{31}{24}y^{6}-\frac{17}{6}y^{5}z-\frac{23}{18}y^{5}w-\frac{11}{6}y^{4}z^{2}-\frac{61}{18}y^{4}zw-\frac{235}{216}y^{4}w^{2}-\frac{17}{9}y^{3}z^{2}w-\frac{58}{27}y^{3}zw^{2}-\frac{20}{27}y^{3}w^{3}-\frac{22}{27}y^{2}z^{2}w^{2}-\frac{8}{9}y^{2}zw^{3}-\frac{61}{216}y^{2}w^{4}-\frac{5}{27}yz^{2}w^{3}-\frac{11}{54}yzw^{4}-\frac{1}{18}yw^{5}-\frac{1}{54}z^{2}w^{4}-\frac{1}{54}zw^{5}-\frac{1}{216}w^{6}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y^{2}+yz+\frac{1}{3}yw+\frac{1}{3}zw$ |
The following modular covers realize this modular curve as a fiber product over $X(1)$.
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.
