Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x v + y w + y u $ |
| $=$ | $x w + x u + x v + z v$ |
| $=$ | $x^{2} - x y - y z$ |
| $=$ | $x w - x s + y w - y u + y s$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ - x^{9} y + 2 x^{8} y^{2} + 5 x^{8} z^{2} + 2 x^{7} y^{3} - 15 x^{7} y z^{2} - 4 x^{6} y^{4} + \cdots + 5 y^{8} z^{2} $ |
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This modular curve has 8 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:0:1/2:0:0:1:0)$, $(0:0:0:1:1:2:3:0:1)$, $(0:0:0:0:0:0:-1:1:0)$, $(0:0:0:0:0:0:0:-1/2:1)$, $(0:0:0:1:1:-1:0:0:1)$, $(0:0:0:-1/2:-1/5:1/2:0:-9/10:1)$, $(0:0:0:0:-2:0:0:1:0)$, $(0:0:0:1:-1/5:2:-3:18/5:1)$ |
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 \frac{3^3}{2^6}\cdot\frac{1043909767478125824wr^{9}+387835473367624448wr^{8}s-114688499141169088wr^{7}s^{2}+167427885853186074wr^{6}s^{3}-10030259806026691564wr^{5}s^{4}+1538144643984433995wr^{4}s^{5}+454238903762500518wr^{3}s^{6}+19300584402562050wr^{2}s^{7}-4180378585027311wrs^{8}-67073101553664ws^{9}+1096482221913783040tvr^{8}+314358137974026048tvr^{7}s-152241380853909312tvr^{6}s^{2}+187658216898076142tvr^{5}s^{3}-10552085534194285380tvr^{4}s^{4}+2514320524262332737tvr^{3}s^{5}+332937080351261010tvr^{2}s^{6}-9131500108369530tvrs^{7}-2173112771903469tvs^{8}-215770311648000tr^{9}-1664785450380866048tr^{8}s-528739145544088624tr^{7}s^{2}+194894534698886796tr^{6}s^{3}-298336045119491592tr^{5}s^{4}+15983431425967227930tr^{4}s^{5}-3368427012009217152tr^{3}s^{6}-472724034863436900tr^{2}s^{7}+12066642222502230trs^{8}+6362865165947220ts^{9}+912113472566185600uvr^{8}+208176978610938624uvr^{7}s-158113746420760200uvr^{6}s^{2}+150766274006854310uvr^{5}s^{3}-8791920322402722192uvr^{4}s^{4}+2588465433982805973uvr^{3}s^{5}+277272438870261474uvr^{2}s^{6}-17078505126595974uvrs^{7}-3598303653575433uvs^{8}-165721719744000ur^{9}-1402259701355170688ur^{8}s-408653819312807792ur^{7}s^{2}+186580966919055354ur^{6}s^{3}-246416975314239884ur^{5}s^{4}+13474389391312036875ur^{4}s^{5}-3179434958184871854ur^{3}s^{6}-406444344921115902ur^{2}s^{7}+16008287177564217urs^{8}+6279774201123156us^{9}-250474166662422784v^{2}r^{8}-53299880897645312v^{2}r^{7}s+45995280289629600v^{2}r^{6}s^{2}-40516209636652710v^{2}r^{5}s^{3}+2417034970618285668v^{2}r^{4}s^{4}-745793724828781269v^{2}r^{3}s^{5}-80480784140881146v^{2}r^{2}s^{6}+4181768205860466v^{2}rs^{7}+1823556617073873v^{2}s^{8}-250362982034745088vr^{9}-803754897821986432vr^{8}s-108271267755792192vr^{7}s^{2}+102686547690631578vr^{6}s^{3}+2298321347994131204vr^{5}s^{4}+6503204550536675019vr^{4}s^{5}-2348627883426495090vr^{3}s^{6}-232329097058587182vr^{2}s^{7}+18248129978324193vrs^{8}+4410171491481144vs^{9}+117481748544000r^{10}+161832550079856896r^{9}s+161570162521764256r^{8}s^{2}+41480888035992360r^{7}s^{3}+36099840468986380r^{6}s^{4}-1517429108697640924r^{5}s^{5}-696139171647582462r^{4}s^{6}+72370214739124236r^{3}s^{7}+21203548355497536r^{2}s^{8}+1418607718097022rs^{9}-11573778493440s^{10}}{7822012142976wr^{9}+5740705122720wr^{8}s+1238163072112wr^{7}s^{2}+1368296842446wr^{6}s^{3}-75827210046364wr^{5}s^{4}-16492424987175wr^{4}s^{5}-1918742083902wr^{3}s^{6}-121853628882wr^{2}s^{7}-557770293wrs^{8}+8214234559232tvr^{8}+5322019655200tvr^{7}s+792106254480tvr^{6}s^{2}+1420679809674tvr^{5}s^{3}-79266650312724tvr^{4}s^{4}-9893029886037tvr^{3}s^{5}-441922040730tvr^{2}s^{6}-6020626806tvrs^{7}-567882495tvs^{8}-12461378788736tr^{8}s-8436174581616tr^{7}s^{2}-2280302540124tr^{6}s^{3}-5648272157224tr^{5}s^{4}+114562893785790tr^{4}s^{5}+14231138947776tr^{3}s^{6}+545750907012tr^{2}s^{7}+13677544818trs^{8}+414535644ts^{9}+6834855109184uvr^{8}+4041105752608uvr^{7}s+311912028600uvr^{6}s^{2}+690195996306uvr^{5}s^{3}-67458618195000uvr^{4}s^{4}-6409807132473uvr^{3}s^{5}-183283240266uvr^{2}s^{6}-10644591834uvrs^{7}-706061043uvs^{8}-10497149906592ur^{8}s-6836126026144ur^{7}s^{2}-1517732666898ur^{6}s^{3}-3766057048700ur^{5}s^{4}+98488122100089ur^{4}s^{5}+11252022514470ur^{3}s^{6}+385463462958ur^{2}s^{7}+13293900387urs^{8}+414535644us^{9}-1876801998208v^{2}r^{8}-1081700291616v^{2}r^{7}s-58509030960v^{2}r^{6}s^{2}-47928490034v^{2}r^{5}s^{3}+18909312406548v^{2}r^{4}s^{4}+1875424002489v^{2}r^{3}s^{5}+79201563426v^{2}r^{2}s^{6}+1877481342v^{2}rs^{7}+429703947v^{2}s^{8}-1876801998208vr^{9}-6712029718624vr^{8}s-3278065388240vr^{7}s^{2}+201708850510vr^{6}s^{3}+20006379610356vr^{5}s^{4}+59738571218457vr^{4}s^{5}+5711995522170vr^{3}s^{6}+180658192158vr^{2}s^{7}+8715994875vrs^{8}+276357096vs^{9}+1204525682176r^{9}s+1619885160736r^{8}s^{2}+1247546296984r^{7}s^{3}+2485843308612r^{6}s^{4}-7548339910596r^{5}s^{5}-5781603085290r^{4}s^{6}-796514329116r^{3}s^{7}-29159435376r^{2}s^{8}-240409782rs^{9}}$ |
Map
of degree 1 from the canonical model of this modular curve to the plane model of the modular curve
60.144.9.ft.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle x$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{5}w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ -X^{9}Y+2X^{8}Y^{2}+5X^{8}Z^{2}+2X^{7}Y^{3}-15X^{7}YZ^{2}-4X^{6}Y^{4}-15X^{6}Y^{2}Z^{2}-2X^{5}Y^{5}+30X^{5}Y^{3}Z^{2}+2X^{4}Y^{6}+40X^{4}Y^{4}Z^{2}+X^{3}Y^{7}-30X^{3}Y^{5}Z^{2}-225X^{3}Y^{3}Z^{4}-15X^{2}Y^{6}Z^{2}+15XY^{7}Z^{2}+225XY^{5}Z^{4}+5Y^{8}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.
