Canonical model in $\mathbb{P}^{ 6 }$ defined by 10 equations
| $ 0 $ | $=$ | $ x z - x u + y t $ |
| $=$ | $ - x w + y^{2} - y w$ |
| $=$ | $y z - y u - z w + w t + w u$ |
| $=$ | $2 x z + x t + x u - x v - w t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 5 x^{6} z^{2} + 20 x^{5} y z^{2} + 9 x^{4} y^{4} + 5 x^{4} y^{2} z^{2} + 25 x^{4} z^{4} + \cdots - 20 y^{6} z^{2} $ |
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This modular curve has 4 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:1/3:0:0:-2/3:1)$, $(0:0:0:0:-2:1:0)$, $(0:0:0:0:0:1:0)$, $(0:0:1/3:0:2:-2/3:1)$ |
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 2^2\,\frac{5925368083ztu^{10}+64006537564ztu^{9}v+262366178467ztu^{8}v^{2}+500843751512ztu^{7}v^{3}+417408022670ztu^{6}v^{4}+52738740600ztu^{5}v^{5}-104045771418ztu^{4}v^{6}-32402553384ztu^{3}v^{7}+4067344911ztu^{2}v^{8}-903759300ztuv^{9}-757085337ztv^{10}+8841925662zu^{11}+102753610345zu^{10}v+442358452771zu^{9}v^{2}+861890764738zu^{8}v^{3}+689112503768zu^{7}v^{4}+17090359322zu^{6}v^{5}-184917144882zu^{5}v^{6}+50436855048zu^{4}v^{7}+121376560074zu^{3}v^{8}+53286439629zu^{2}v^{9}+11035349343zuv^{10}+736933014zv^{11}-14527817370w^{2}u^{10}-125993155320w^{2}u^{9}v-391301723070w^{2}u^{8}v^{2}-504588225600w^{2}u^{7}v^{3}-177614474220w^{2}u^{6}v^{4}+132060661200w^{2}u^{5}v^{5}+91499458140w^{2}u^{4}v^{6}+13549343040w^{2}u^{3}v^{7}+11459168550w^{2}u^{2}v^{8}+5361987240w^{2}uv^{9}+220568130w^{2}v^{10}+2954806537t^{2}u^{10}+30673402660t^{2}u^{9}v+121202332645t^{2}u^{8}v^{2}+224183064920t^{2}u^{7}v^{3}+182225003810t^{2}u^{6}v^{4}+24445501512t^{2}u^{5}v^{5}-36025947870t^{2}u^{4}v^{6}+253014360t^{2}u^{3}v^{7}+13493070885t^{2}u^{2}v^{8}+4454607780t^{2}uv^{9}+422190153t^{2}v^{10}+5885614610tu^{11}+61242568009tu^{10}v+233106165142tu^{9}v^{2}+378550314321tu^{8}v^{3}+177058131116tu^{7}v^{4}-155546540390tu^{6}v^{5}-131884445700tu^{5}v^{6}+27478207746tu^{4}v^{7}+25566193458tu^{3}v^{8}-6879250227tu^{2}v^{9}-4283116290tuv^{10}-418306851tv^{11}+3072u^{12}-23721652u^{11}v-12083676247u^{10}v^{2}-89876452063u^{9}v^{3}-231909929546u^{8}v^{4}-230517120020u^{7}v^{5}-31497099606u^{6}v^{6}+70494786690u^{5}v^{7}+24000886416u^{4}v^{8}-1533434328u^{3}v^{9}+1492314237u^{2}v^{10}+830581965uv^{11}+44116698v^{12}}{6750ztu^{10}+10800ztu^{9}v-41454ztu^{8}v^{2}+68832ztu^{7}v^{3}-45964ztu^{6}v^{4}-501056ztu^{5}v^{5}+230860ztu^{4}v^{6}+95328ztu^{3}v^{7}-551154ztu^{2}v^{8}+110288ztuv^{9}+131522ztv^{10}+30375zu^{11}-6075zu^{10}v+484821zu^{9}v^{2}+5073039zu^{8}v^{3}+11691090zu^{7}v^{4}+5536166zu^{6}v^{5}-6409226zu^{5}v^{6}-3303182zu^{4}v^{7}+1671927zu^{3}v^{8}+294885zu^{2}v^{9}-130747zuv^{10}-1873zv^{11}+16875w^{2}u^{10}-9000w^{2}u^{9}v+279345w^{2}u^{8}v^{2}+2432880w^{2}u^{7}v^{3}+3803370w^{2}u^{6}v^{4}-1056240w^{2}u^{5}v^{5}-2476050w^{2}u^{4}v^{6}+854160w^{2}u^{3}v^{7}+21915w^{2}u^{2}v^{8}-276840w^{2}uv^{9}+78705w^{2}v^{10}-3375t^{2}u^{10}+16200t^{2}u^{9}v+131715t^{2}u^{8}v^{2}+834000t^{2}u^{7}v^{3}+1628894t^{2}u^{6}v^{4}+273712t^{2}u^{5}v^{5}-993110t^{2}u^{4}v^{6}-81360t^{2}u^{3}v^{7}+24225t^{2}u^{2}v^{8}-1528t^{2}uv^{9}+57731t^{2}v^{10}-6750tu^{11}+45900tu^{10}v+255330tu^{9}v^{2}+1390272tu^{8}v^{3}+1951276tu^{7}v^{4}-1236312tu^{6}v^{5}-1525844tu^{5}v^{6}+1048640tu^{4}v^{7}-139374tu^{3}v^{8}-25652tu^{2}v^{9}+109394tuv^{10}-73472tv^{11}+40500u^{11}v-48825u^{10}v^{2}-70596u^{9}v^{3}+867501u^{8}v^{4}+1790680u^{7}v^{5}+50978u^{6}v^{6}-1334248u^{5}v^{7}+36582u^{4}v^{8}+426228u^{3}v^{9}-37625u^{2}v^{10}-65876uv^{11}+15741v^{12}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
60.144.7.lv.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{3}{5}z$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 5X^{6}Z^{2}+20X^{5}YZ^{2}+9X^{4}Y^{4}+5X^{4}Y^{2}Z^{2}+25X^{4}Z^{4}+36X^{3}Y^{5}-80X^{3}Y^{3}Z^{2}+100X^{3}YZ^{4}+45X^{2}Y^{6}-125X^{2}Y^{4}Z^{2}+125X^{2}Y^{2}Z^{4}+18XY^{7}-70XY^{5}Z^{2}+50XY^{3}Z^{4}-20Y^{6}Z^{2} $ |
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.
