Canonical model in $\mathbb{P}^{ 4 }$ defined by 3 equations
| $ 0 $ | $=$ | $ x^{2} - x y + y^{2} - y w - z^{2} - z w - w^{2} $ |
| $=$ | $x^{2} - x y + y z + y w + z^{2} + z w + w^{2}$ |
| $=$ | $x^{2} - x y - 5 y z + y w - 2 z^{2} + z w + w^{2} + 2 t^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 240 x^{8} - 96 x^{7} y - 960 x^{7} z + 64 x^{6} y^{2} + 384 x^{6} y z + 2256 x^{6} z^{2} - 16 x^{5} y^{3} + \cdots + 321 z^{8} $ |
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This modular curve has no real points and no $\Q_p$ points for $p=23$, and therefore no rational points.
Maps to other modular curves
$j$-invariant map
of degree 192 from the canonical model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^8}{3^3\cdot5^4}\cdot\frac{2670392498874664306640625000yw^{23}+4955347040732127701234558622yw^{21}t^{2}+4075772035323375138834459900yw^{19}t^{4}+1950382510523348147790342300yw^{17}t^{6}+598950246241936159661826000yw^{15}t^{8}+122740133158603961141287500yw^{13}t^{10}+16900041373286449209075000yw^{11}t^{12}+1534166743186204801875000yw^{9}t^{14}+87494824175667260625000yw^{7}t^{16}+2856790468674724218750yw^{5}t^{18}+44541287200898437500yw^{3}t^{20}+214230725507812500ywt^{22}+1221789365828434871819602524z^{2}w^{22}+2165412846349549364167244655z^{2}w^{20}t^{2}+1689946083978833799395176500z^{2}w^{18}t^{4}+760944121591470313024064250z^{2}w^{16}t^{6}+217502002501677765655650000z^{2}w^{14}t^{8}+40882771641595925114493750z^{2}w^{12}t^{10}+5058851999573500033781250z^{2}w^{10}t^{12}+400556181941069934375000z^{2}w^{8}t^{14}+19015684310768723437500z^{2}w^{6}t^{16}+476736674801396484375z^{2}w^{4}t^{18}+4802840905605468750z^{2}w^{2}t^{20}+5596371386718750z^{2}t^{22}+2312627742103205472280148268zw^{23}+4359072446105937400520906922zw^{21}t^{2}+3641619201263716539032693100zw^{19}t^{4}+1769836419564815242359366300zw^{17}t^{6}+551918796358443350053851000zw^{15}t^{8}+114827564514766847246212500zw^{13}t^{10}+16045777389836518584075000zw^{11}t^{12}+1477362133905239694375000zw^{9}t^{14}+85366246452363649687500zw^{7}t^{16}+2819378823239060156250zw^{5}t^{18}+44363128533398437500zw^{3}t^{20}+214230725507812500zwt^{22}+1954862985327243804931640625w^{24}+4235536468209778716183226506w^{22}t^{2}+4078409276406453753665941470w^{20}t^{4}+2296730272637872517884366500w^{18}t^{6}+836923655445966190978356375w^{16}t^{8}+206017157357086786125412500w^{14}t^{10}+34683691757509294796475000w^{12}t^{12}+3950697171007943703562500w^{10}t^{14}+293985531174670013671875w^{8}t^{16}+13340060830038803906250w^{6}t^{18}+324502085353535156250w^{4}t^{20}+3237965995078125000w^{2}t^{22}+6115100048828125t^{24}}{t^{4}(72596966364yw^{17}t^{2}+477646573680yw^{15}t^{4}+204644515500yw^{13}t^{6}+5020861282803000yw^{11}t^{8}+4296367610100000yw^{9}t^{10}+1372232879100000yw^{7}t^{12}+196463307000000yw^{5}t^{14}+11668650000000yw^{3}t^{16}+192600000000ywt^{18}-718906104162z^{2}w^{18}-738559500930z^{2}w^{16}t^{2}+2221219989000z^{2}w^{14}t^{4}+1356657153750z^{2}w^{12}t^{6}+2296911957648750z^{2}w^{10}t^{8}+1774326243262500z^{2}w^{8}t^{10}+490516887375000z^{2}w^{6}t^{12}+56170762500000z^{2}w^{4}t^{14}+2239312500000z^{2}w^{2}t^{16}+13500000000z^{2}t^{18}-435581798184zw^{19}-3141813293316zw^{17}t^{2}-1744330117320zw^{15}t^{4}+1757307352500zw^{13}t^{6}+4348748607453000zw^{11}t^{8}+3847740880500000zw^{9}t^{10}+1271690522100000zw^{7}t^{12}+188199807000000zw^{5}t^{14}+11494050000000zw^{3}t^{16}+192600000000zwt^{18}-145193932728w^{18}t^{2}-927453413745w^{16}t^{4}-447464209500w^{14}t^{6}+3676045537305000w^{12}t^{8}+4288304220682500w^{10}t^{10}+1919776266675000w^{8}t^{12}+408242658250000w^{6}t^{14}+40555325000000w^{4}t^{16}+1510875000000w^{2}t^{18}+9000000000t^{20})}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
24.192.5.bx.2
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y-2z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 6x+6t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle 2w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 240X^{8}-96X^{7}Y+64X^{6}Y^{2}-16X^{5}Y^{3}+4X^{4}Y^{4}-960X^{7}Z+384X^{6}YZ-256X^{5}Y^{2}Z+64X^{4}Y^{3}Z-16X^{3}Y^{4}Z+2256X^{6}Z^{2}-912X^{5}YZ^{2}+536X^{4}Y^{2}Z^{2}-112X^{3}Y^{3}Z^{2}+24X^{2}Y^{4}Z^{2}-3408X^{5}Z^{3}+1392X^{4}YZ^{3}-712X^{3}Y^{2}Z^{3}+112X^{2}Y^{3}Z^{3}-16XY^{4}Z^{3}+2868X^{4}Z^{4}-1272X^{3}YZ^{4}+500X^{2}Y^{2}Z^{4}-64XY^{3}Z^{4}+4Y^{4}Z^{4}-1176X^{3}Z^{5}+672X^{2}YZ^{5}-112XY^{2}Z^{5}+16Y^{3}Z^{5}-624X^{2}Z^{6}-96XYZ^{6}-20Y^{2}Z^{6}+804XZ^{7}-72YZ^{7}+321Z^{8} $ |
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.
