Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} t - x z t + x w t + y^{2} t + y z t $ |
| $=$ | $x^{2} w - x z w + x w^{2} + y^{2} w + y z w$ |
| $=$ | $x^{2} z - x z^{2} + x z w + y^{2} z + y z^{2}$ |
| $=$ | $x^{2} t + 2 x z t + x w t - y^{2} t + z^{2} t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{7} - 4 x^{6} z + x^{5} y z + 15 x^{5} z^{2} - 5 x^{4} y z^{2} - 14 x^{4} z^{3} - x^{3} y^{2} z^{2} + \cdots - 136 z^{7} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ 5x^{7} + 25x^{6} + 35x^{5} + 50x^{4} + 35x^{3} + 25x^{2} + 5x $ |
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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}{5^2}\cdot\frac{158203125000xz^{13}+537500000000xz^{11}t^{2}-25000000000xz^{10}t^{3}-704765625000xz^{9}t^{4}+1001675000000xz^{8}t^{5}+125335000000xz^{7}t^{6}-1708950000000xz^{6}t^{7}+963803150000xz^{5}t^{8}+2494987600000xz^{4}t^{9}-4599695899000xz^{3}t^{10}+721249967200xz^{2}t^{11}+8947249732680xzt^{12}-166994527230xw^{13}+1521507557520xw^{12}t-2867215350800xw^{11}t^{2}-9405872351040xw^{10}t^{3}+38434347742560xw^{9}t^{4}-19069889600000xw^{8}t^{5}-83453859612160xw^{7}t^{6}+94150800773120xw^{6}t^{7}+75895294645760xw^{5}t^{8}-129303297669120xw^{4}t^{9}-30369336854080xw^{3}t^{10}+82440535990400xw^{2}t^{11}+9107517493760xwt^{12}-14473996466240xt^{13}+158203125000yz^{13}-157812500000yz^{11}t^{2}+278125000000yz^{10}t^{3}-168359375000yz^{9}t^{4}-266725000000yz^{8}t^{5}+254215000000yz^{7}t^{6}+775944000000yz^{6}t^{7}-1242212600000yz^{5}t^{8}-1153160400000yz^{4}t^{9}+4563402695000yz^{3}t^{10}-2133562922400yz^{2}t^{11}-8511464906680yzt^{12}+86262099066yw^{13}-829024375392yw^{12}t+1848694339520yw^{11}t^{2}+4548750450240yw^{10}t^{3}-23097866345440yw^{9}t^{4}+16394989800960yw^{8}t^{5}+49494219788800yw^{7}t^{6}-72080635146240yw^{6}t^{7}-45628704345600yw^{5}t^{8}+103019872808960yw^{4}t^{9}+16571592133120yw^{3}t^{10}-76361739182080yw^{2}t^{11}-6525274042880ywt^{12}+14946043393024yt^{13}+158203125000z^{14}-158593750000z^{12}t^{2}+278750000000z^{11}t^{3}-628984375000z^{10}t^{4}+317475000000z^{9}t^{5}+433850000000z^{8}t^{6}-947070000000z^{7}t^{7}+391829600000z^{6}t^{8}+1511754800000z^{5}t^{9}-3713414549000z^{4}t^{10}+2241825170400z^{3}t^{11}+6734327246600z^{2}t^{12}-205849934010zw^{13}+1809501191280zw^{12}t-3077996882800zw^{11}t^{2}-11573792781760zw^{10}t^{3}+42789650487840zw^{9}t^{4}-17044036968960zw^{8}t^{5}-91917210324480zw^{7}t^{6}+95031464448000zw^{6}t^{7}+82709532076800zw^{5}t^{8}-131984056057600zw^{4}t^{9}-29698859289600zw^{3}t^{10}+83131370366400zw^{2}t^{11}+8308375387200zwt^{12}-14378809376960zt^{13}-109957652847w^{14}+985045845744w^{13}t-1736307922240w^{12}t^{2}-6288750292320w^{11}t^{3}+23934642002640w^{10}t^{4}-9893416748800w^{9}t^{5}-51999096070400w^{8}t^{6}+52961174661120w^{7}t^{7}+46960108162560w^{6}t^{8}-71665806059520w^{5}t^{9}-18004502905920w^{4}t^{10}+42121223422080w^{3}t^{11}+4610522315600w^{2}t^{12}-6977178347408wt^{13}-50000t^{14}}{625000xz^{7}t^{6}-1500000xz^{6}t^{7}+3950000xz^{5}t^{8}-7000000xz^{4}t^{9}+8404000xz^{3}t^{10}-3963200xz^{2}t^{11}-8376320xzt^{12}+1773xw^{13}+7890xw^{12}t-40755xw^{11}t^{2}-329930xw^{10}t^{3}-603615xw^{9}t^{4}+597304xw^{8}t^{5}+2298308xw^{7}t^{6}+12038360xw^{6}t^{7}-25400730xw^{5}t^{8}+24228960xw^{4}t^{9}-16016608xw^{3}t^{10}-40684416xw^{2}t^{11}+29053120xwt^{12}+23279616xt^{13}-625000yz^{7}t^{6}+1500000yz^{6}t^{7}-4200000yz^{5}t^{8}+7600000yz^{4}t^{9}-9884000yz^{3}t^{10}+6739200yz^{2}t^{11}+4152320yzt^{12}-5868yw^{13}-6210yw^{12}t+126540yw^{11}t^{2}+244470yw^{10}t^{3}-623710yw^{9}t^{4}-1501274yw^{8}t^{5}+1884222yw^{7}t^{6}-872040yw^{6}t^{7}+11174470yw^{5}t^{8}-25465600yw^{4}t^{9}+7871168yw^{3}t^{10}+35052416yw^{2}t^{11}-17502400ywt^{12}-17488896yt^{13}+625000z^{8}t^{6}-1500000z^{7}t^{7}+4200000z^{6}t^{8}-7800000z^{5}t^{9}+10414000z^{4}t^{10}-7963200z^{3}t^{11}-2629120z^{2}t^{12}+423zw^{13}-810zw^{12}t-44505zw^{11}t^{2}-266430zw^{10}t^{3}-662365zw^{9}t^{4}-434296zw^{8}t^{5}+336108zw^{7}t^{6}+15406600zw^{6}t^{7}-19545870zw^{5}t^{8}+35115360zw^{4}t^{9}-24224608zw^{3}t^{10}-50603136zw^{2}t^{11}+14086720zwt^{12}+14585856zt^{13}-1881w^{14}+315w^{13}t+24975w^{12}t^{2}-134205w^{11}t^{3}-779420w^{10}t^{4}-376803w^{9}t^{5}+2425029w^{8}t^{6}+10520540w^{7}t^{7}-16144705w^{6}t^{8}+11507600w^{5}t^{9}-13053264w^{4}t^{10}-21194048w^{3}t^{11}+17776160w^{2}t^{12}+12361728wt^{13}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
60.96.3.p.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 3t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{7}-4X^{6}Z+X^{5}YZ+15X^{5}Z^{2}-5X^{4}YZ^{2}-X^{3}Y^{2}Z^{2}-14X^{4}Z^{3}+4X^{3}YZ^{3}+6X^{2}Y^{2}Z^{3}-49X^{3}Z^{4}+8X^{2}YZ^{4}-9XY^{2}Z^{4}+120X^{2}Z^{5}-40XYZ^{5}+4Y^{2}Z^{5}-176XZ^{6}+32YZ^{6}-136Z^{7} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
60.96.3.p.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{1}{3}y+\frac{2}{3}z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{2}{27}y^{4}-\frac{4}{27}y^{3}z-\frac{2}{9}y^{2}zt+\frac{2}{27}yz^{3}+\frac{5}{9}yz^{2}t-\frac{4}{27}z^{4}-\frac{2}{9}z^{3}t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{1}{3}y-\frac{1}{3}z$ |
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.
