Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ 2 x^{2} t - x z t + 2 x w t + y z t $ |
| $=$ | $x^{2} t + x z t + x w t - y^{2} t + 2 y z t$ |
| $=$ | $2 x^{2} z + x^{2} w - x z^{2} + x w^{2} + y^{2} w + y z^{2} - y z w$ |
| $=$ | $x^{2} z + x z^{2} + x z w - y^{2} z + 2 y z^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 36 x^{7} - 108 x^{6} z + 144 x^{5} z^{2} - 108 x^{4} z^{3} - 60 x^{3} y^{2} z^{2} + 47 x^{3} z^{4} + \cdots + 15 y^{2} z^{5} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 15x^{7} + 75x^{6} + 105x^{5} + 150x^{4} + 105x^{3} + 75x^{2} + 15x $ |
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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{1}{2^3\cdot5^2}\cdot\frac{7695701885399040xzw^{12}-350993633136583680xzw^{10}t^{2}-94569923379565440xzw^{8}t^{4}-2700840502457280xzw^{6}t^{6}+135957971140200xzw^{4}t^{8}-5831031216690xzw^{2}t^{10}-188907269120xzt^{12}+138104298114600960xw^{13}+203762589244609536xw^{11}t^{2}+11242041176689920xw^{9}t^{4}-3155915716496064xw^{7}t^{6}+22786256331864xw^{5}t^{8}+7712271547800xw^{3}t^{10}-181256278016xwt^{12}-285119385126174720yzw^{12}-111328038391311360yzw^{10}t^{2}+24796056836257920yzw^{8}t^{4}+6379499338871040yzw^{6}t^{6}+346516615841400yzw^{4}t^{8}+13752311774670yzw^{2}t^{10}+275010319360yzt^{12}+139319385126174720yw^{13}-11589458708302848yw^{11}t^{2}-23227644267244800yw^{9}t^{4}-1014124681081728yw^{7}t^{6}+85453273344408yw^{5}t^{8}+1757112149400yw^{3}t^{10}+252436697088ywt^{12}+168883726746562560z^{2}w^{12}+164714095586856960z^{2}w^{10}t^{2}+13129141445507520z^{2}w^{8}t^{4}-2796368648417280z^{2}w^{6}t^{6}-400900318587000z^{2}w^{4}t^{8}-24725095317585z^{2}w^{2}t^{10}-488247906560z^{2}t^{12}-368143068366950400zw^{13}-328658411580106752zw^{11}t^{2}-5337100416435840zw^{9}t^{4}+7339014714938688zw^{7}t^{6}+309599069735232zw^{5}t^{8}-3768293820870zw^{3}t^{10}-516815922688zwt^{12}+195614341620387840w^{14}+145678856984838144w^{12}t^{2}-10469445608900160w^{10}t^{4}-3207130244732736w^{8}t^{6}+111580458426216w^{6}t^{8}+7914908941875w^{4}t^{10}-75542813189w^{2}t^{12}}{t^{4}(5055670080xzw^{8}+3081682260xzw^{6}t^{2}+423049725xzw^{4}t^{4}+11706870xzw^{2}t^{6}+29385xzt^{8}-2524420080xw^{9}-1183162248xw^{7}t^{2}-49575222xw^{5}t^{4}+7280460xw^{3}t^{6}+135402xwt^{8}-5058145440yzw^{8}-3251959380yzw^{6}t^{2}-485028225yzw^{4}t^{4}-15160710yzw^{2}t^{6}-45405yzt^{8}+2526895440yw^{9}+1268774064yw^{7}t^{2}+68276916yw^{5}t^{4}-8145360yw^{3}t^{6}-186276ywt^{8}+8856258120z^{2}w^{8}+5678006850z^{2}w^{6}t^{2}+846508950z^{2}w^{4}t^{4}+26569740z^{2}w^{2}t^{6}+81630z^{2}t^{8}-2529370800zw^{9}-1143375804zw^{7}t^{2}+9317664zw^{5}t^{4}+17018280zw^{3}t^{6}+335616zwt^{8}-1264387320w^{10}-568715742w^{8}t^{2}-20746818w^{6}t^{4}+3426360w^{4}t^{6}+57268w^{2}t^{8})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
60.96.3.q.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{10}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 3z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 36X^{7}-108X^{6}Z+144X^{5}Z^{2}-60X^{3}Y^{2}Z^{2}-108X^{4}Z^{3}+120X^{2}Y^{2}Z^{3}+47X^{3}Z^{4}-75XY^{2}Z^{4}-11X^{2}Z^{5}+15Y^{2}Z^{5}+XZ^{6} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
60.96.3.q.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{1}{3}y+z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{1}{9}y^{2}zt+\frac{1}{2}yz^{2}t-\frac{1}{2}z^{3}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{1}{3}y$ |
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.
