Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x^{2} + x y + y z $ |
| $=$ | $7 x^{2} - 8 x y + 5 y^{2} + 2 y z + 5 z^{2} + w^{2} + t^{2}$ |
| $=$ | $3 x^{2} - 2 x y + 10 x z - 5 y^{2} - 2 y z + 2 w t - t^{2}$ |
![Copy content]()
![Toggle raw display]()
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{8} + 10 x^{6} y^{2} - 20 x^{5} y^{3} + 5 x^{4} y^{4} + 6 x^{4} y^{2} z^{2} - 20 x^{3} y^{5} + \cdots + y^{4} z^{4} $ |
![Copy content]()
![Toggle raw display]()
This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map
of degree 144 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2\cdot3^3\,\frac{37150xzw^{16}+508800xzw^{15}t+4938800xzw^{14}t^{2}+29692400xzw^{13}t^{3}+97943800xzw^{12}t^{4}+194106400xzw^{11}t^{5}+205470000xzw^{10}t^{6}-144098800xzw^{9}t^{7}-746703900xzw^{8}t^{8}-1022404800xzw^{7}t^{9}-637705200xzw^{6}t^{10}+38480400xzw^{5}t^{11}+266922200xzw^{4}t^{12}+103437600xzw^{3}t^{13}-21714800xzw^{2}t^{14}-16438800xzwt^{15}+3099950xzt^{16}-57850yzw^{16}-488400yzw^{15}t-1730000yzw^{14}t^{2}-4343200yzw^{13}t^{3}+5245800yzw^{12}t^{4}+99434000yzw^{11}t^{5}+320482000yzw^{10}t^{6}+567576000yzw^{9}t^{7}+591941700yzw^{8}t^{8}+200645200yzw^{7}t^{9}-222378800yzw^{6}t^{10}-292087200yzw^{5}t^{11}-111383000yzw^{4}t^{12}+20921200yzw^{3}t^{13}+20337200yzw^{2}t^{14}+1673600yzwt^{15}-1125850yzt^{16}-29925z^{2}w^{16}-169200z^{2}w^{15}t+18600z^{2}w^{14}t^{2}-124800z^{2}w^{13}t^{3}-9563600z^{2}w^{12}t^{4}-28924400z^{2}w^{11}t^{5}-46328000z^{2}w^{10}t^{6}-56345600z^{2}w^{9}t^{7}+24824550z^{2}w^{8}t^{8}+112983600z^{2}w^{7}t^{9}+88237000z^{2}w^{6}t^{10}+5382400z^{2}w^{5}t^{11}-48470600z^{2}w^{4}t^{12}-18469200z^{2}w^{3}t^{13}+7931600z^{2}w^{2}t^{14}+4227200z^{2}wt^{15}-1112425z^{2}t^{16}-6561w^{18}-29866w^{17}t+111348w^{16}t^{2}+961392w^{15}t^{3}+3207780w^{14}t^{4}+8896864w^{13}t^{5}+13046124w^{12}t^{6}-10064832w^{11}t^{7}-68298558w^{10}t^{8}-114602580w^{9}t^{9}-85085812w^{8}t^{10}+16631088w^{7}t^{11}+72242196w^{6}t^{12}+32690864w^{5}t^{13}-14228780w^{4}t^{14}-13913568w^{3}t^{15}+977127w^{2}t^{16}+2294254wt^{17}-419904t^{18}}{680xzw^{16}+4680xzw^{15}t+14770xzw^{14}t^{2}+13460xzw^{13}t^{3}-19330xzw^{12}t^{4}-96140xzw^{11}t^{5}-69660xzw^{10}t^{6}-89140xzw^{9}t^{7}+115560xzw^{8}t^{8}-21340xzw^{7}t^{9}-56430xzw^{6}t^{10}+121840xzw^{5}t^{11}-82310xzw^{4}t^{12}+24080xzw^{3}t^{13}-1360xzw^{2}t^{14}-960xzwt^{15}+160xzt^{16}-220yzw^{16}-2030yzw^{15}t-10570yzw^{14}t^{2}-27680yzw^{13}t^{3}-44350yzw^{12}t^{4}-12250yzw^{11}t^{5}+5120yzw^{10}t^{6}+116060yzw^{9}t^{7}-104580yzw^{8}t^{8}+243330yzw^{7}t^{9}-319090yzw^{6}t^{10}+276140yzw^{5}t^{11}-177450yzw^{4}t^{12}+77110yzw^{3}t^{13}-21740yzw^{2}t^{14}+3560yzwt^{15}-240yzt^{16}-230z^{2}w^{15}t-635z^{2}w^{14}t^{2}+265z^{2}w^{13}t^{3}+7125z^{2}w^{12}t^{4}+10505z^{2}w^{11}t^{5}+9440z^{2}w^{10}t^{6}-7510z^{2}w^{9}t^{7}-26820z^{2}w^{8}t^{8}+102960z^{2}w^{7}t^{9}-219895z^{2}w^{6}t^{10}+243905z^{2}w^{5}t^{11}-154405z^{2}w^{4}t^{12}+58525z^{2}w^{3}t^{13}-12730z^{2}w^{2}t^{14}+1340z^{2}wt^{15}-40z^{2}t^{16}+90w^{17}t+717w^{16}t^{2}+2481w^{15}t^{3}+3167w^{14}t^{4}-1383w^{13}t^{5}-10878w^{12}t^{6}-8958w^{11}t^{7}-2528w^{10}t^{8}+5472w^{9}t^{9}+19037w^{8}t^{10}-40419w^{7}t^{11}+47421w^{6}t^{12}-34779w^{5}t^{13}+15856w^{4}t^{14}-4488w^{3}t^{15}+720w^{2}t^{16}-48wt^{17}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
60.144.5.oc.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 5X^{8}+10X^{6}Y^{2}-20X^{5}Y^{3}+5X^{4}Y^{4}+6X^{4}Y^{2}Z^{2}-20X^{3}Y^{5}+8X^{3}Y^{3}Z^{2}+20X^{2}Y^{6}+10X^{2}Y^{4}Z^{2}-12XY^{5}Z^{2}+4Y^{6}Z^{2}+Y^{4}Z^{4} $ |
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.
