Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ x y + x w + y z $ |
| $=$ | $6 x^{2} - y^{2} - 2 y w - 3 z^{2} + w^{2}$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} + 4 x^{3} z - 3 x^{2} y^{2} + 4 x^{2} z^{2} + 6 x y^{2} z + 3 y^{2} z^{2} - z^{4} $ |
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This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
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}{3}\cdot\frac{14731544520xz^{23}-45460171728xz^{21}w^{2}+29558512224xz^{19}w^{4}-40831045056xz^{17}w^{6}+146040931584xz^{15}w^{8}+31730504855040xz^{13}w^{10}+8808022010880xz^{11}w^{12}-7936962951168xz^{9}w^{14}+1018291709952xz^{7}w^{16}+25676476416xz^{5}w^{18}-4927463424xz^{3}w^{20}-170311680xzw^{22}+1438433640y^{2}z^{22}-4268534112y^{2}z^{20}w^{2}-1080203040y^{2}z^{18}w^{4}+208619224704y^{2}z^{16}w^{6}-3258023378688y^{2}z^{14}w^{8}-18165704702976y^{2}z^{12}w^{10}+2861161159680y^{2}z^{10}w^{12}+1242821283840y^{2}z^{8}w^{14}-238599862272y^{2}z^{6}w^{16}-2271928320y^{2}z^{4}w^{18}+1307271168y^{2}z^{2}w^{20}+40140800y^{2}w^{22}+3711583944yz^{22}w-9926845488yz^{20}w^{3}-33023979936yz^{18}w^{5}-121634851392yz^{16}w^{7}+20714654299392yz^{14}w^{9}+4339495088640yz^{12}w^{11}-7872445043712yz^{10}w^{13}+1946625177600yz^{8}w^{15}-116297275392yz^{6}w^{17}-7171338240yz^{4}w^{19}+558686208yz^{2}w^{21}+23511040yw^{23}+10416775041z^{24}-32724011016z^{22}w^{2}+20410641144z^{20}w^{4}+71257498848z^{18}w^{6}-279454195152z^{16}w^{8}-13239129141504z^{14}w^{10}+3352890703104z^{12}w^{12}+2521791138816z^{10}w^{14}-1010918009088z^{8}w^{16}+91286673408z^{6}w^{18}+3449862144z^{4}w^{20}-455712768z^{2}w^{22}-16625664w^{24}}{w^{2}z^{2}(14722884xz^{17}w^{2}-108965088xz^{15}w^{4}-666247680xz^{13}w^{6}-2600325504xz^{11}w^{8}-3608841600xz^{9}w^{10}-2102340096xz^{7}w^{12}-469628928xz^{5}w^{14}-31721472xz^{3}w^{16}-3072xzw^{18}+19683y^{2}z^{18}+5471874y^{2}z^{16}w^{2}+88494768y^{2}z^{14}w^{4}+631232352y^{2}z^{12}w^{6}+1592641440y^{2}z^{10}w^{8}+1696054464y^{2}z^{8}w^{10}+764944128y^{2}z^{6}w^{12}+137000448y^{2}z^{4}w^{14}+7502592y^{2}z^{2}w^{16}+512y^{2}w^{18}-708588yz^{18}w-34720812yz^{16}w^{3}-452446560yz^{14}w^{5}-1615230720yz^{12}w^{7}-2206953216yz^{10}w^{9}-1023684480yz^{8}w^{11}-56498688yz^{6}w^{13}+42448896yz^{4}w^{15}+4365312yz^{2}w^{17}+1024yw^{19}+12006630z^{18}w^{2}-223074z^{16}w^{4}+333193824z^{14}w^{6}+902630304z^{12}w^{8}+1025234496z^{10}w^{10}+273575232z^{8}w^{12}-91556352z^{6}w^{14}-41080320z^{4}w^{16}-3095040z^{2}w^{18}-512w^{20})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
48.96.1.bg.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{4}-3X^{2}Y^{2}+4X^{3}Z+6XY^{2}Z+4X^{2}Z^{2}+3Y^{2}Z^{2}-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.
