Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} z - 2 x^{2} w + x y z + 2 x y w - z^{2} w - z w^{2} $ |
| $=$ | $ - x^{2} w + y^{2} z + y^{2} w - z^{3} - 2 z^{2} w - z w^{2} - z t^{2}$ |
| $=$ | $x^{2} w + x y w + x w t + y^{2} z + y w t - z^{3} - z^{2} w - z t^{2}$ |
| $=$ | $2 x y z + x y w + y^{2} z + y^{2} w - y z t$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{7} + 23 x^{6} z + 169 x^{5} z^{2} + 17 x^{4} y^{2} z + 441 x^{4} z^{3} + 144 x^{3} y^{2} z^{2} + \cdots + 16 z^{7} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{4} + 1\right) y $ | $=$ | $ 2x^{6} + 7x^{4} + 2x^{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.
| Embedded model |
|---|
| $(0:0:-1:1:0)$, $(1:3:0:0:1)$, $(0:1:0:0:1)$, $(1:-1:0:0:1)$ |
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle 2\,\frac{15006979xw^{10}+280609126xw^{8}t^{2}-45915878xw^{6}t^{4}-2931348xw^{4}t^{6}+6695052xw^{2}t^{8}+1397368xt^{10}-59858902y^{2}w^{8}t+25599940y^{2}w^{6}t^{3}-1002213y^{2}w^{4}t^{5}-3573768y^{2}w^{2}t^{7}-699235y^{2}t^{9}-276079385yzw^{9}+123295194yzw^{7}t^{2}+24753018yzw^{5}t^{4}+526128yzw^{3}t^{6}-80688yzwt^{8}-212664967yw^{10}+182048454yw^{8}t^{2}-73016654yw^{6}t^{4}-9581442yw^{4}t^{6}+2439228yw^{2}t^{8}+698566yt^{10}+105784684z^{2}w^{8}t+65389728z^{2}w^{6}t^{3}+9494760z^{2}w^{4}t^{5}+3765828z^{2}w^{2}t^{7}+699880z^{2}t^{9}+103338067zw^{9}t+115645178zw^{7}t^{3}+10193865zw^{5}t^{5}+4129032zw^{3}t^{7}+783249zwt^{9}-17444848w^{10}t+49214314w^{8}t^{3}+74557019w^{6}t^{5}+8251887w^{4}t^{7}+354713w^{2}t^{9}+689t^{11}}{w^{2}(13040xw^{8}+189180xw^{6}t^{2}-58026xw^{4}t^{4}-17710xw^{2}t^{6}-224xt^{8}-54232y^{2}w^{6}t+35015y^{2}w^{4}t^{3}+5721y^{2}w^{2}t^{5}+60y^{2}t^{7}-239832yzw^{7}+145212yzw^{5}t^{2}+33402yzw^{3}t^{4}+622yzwt^{6}-184752yw^{8}+227420yw^{6}t^{2}-90804yw^{4}t^{4}-13684yw^{2}t^{6}-128yt^{8}+14288z^{2}w^{6}t+84332z^{2}w^{4}t^{3}+3828z^{2}w^{2}t^{5}+8z^{2}t^{7}-30968zw^{7}t+162287zw^{5}t^{3}+1827zw^{3}t^{5}-306zwt^{7}-58296w^{8}t+28287w^{6}t^{3}+95160w^{4}t^{5}+8645w^{2}t^{7}+68t^{9})}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
24.72.3.po.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Equation of the image curve:
| $0$ |
$=$ |
$ X^{7}+23X^{6}Z+17X^{4}Y^{2}Z+169X^{5}Z^{2}+144X^{3}Y^{2}Z^{2}+16XY^{4}Z^{2}+441X^{4}Z^{3}+244X^{2}Y^{2}Z^{3}+16Y^{4}Z^{3}+551X^{3}Z^{4}+152XY^{2}Z^{4}+361X^{2}Z^{5}+32Y^{2}Z^{5}+120XZ^{6}+16Z^{7} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
24.72.3.po.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{1}{16}z^{5}w^{2}-\frac{187}{240}z^{4}w^{3}-\frac{89}{48}z^{3}w^{4}-\frac{217}{120}z^{2}w^{5}-\frac{4}{5}zw^{6}-\frac{2}{15}w^{7}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{983040}z^{22}w^{6}+\frac{83}{1228800}z^{21}w^{7}+\frac{143839}{73728000}z^{20}w^{8}+\frac{13412231}{414720000}z^{19}w^{9}+\frac{1}{61440}z^{19}w^{7}t^{2}+\frac{1137739487}{3317760000}z^{18}w^{10}+\frac{257}{307200}z^{18}w^{8}t^{2}+\frac{1022764549}{414720000}z^{17}w^{11}+\frac{21091}{1152000}z^{17}w^{9}t^{2}+\frac{13851826511}{1105920000}z^{16}w^{12}+\frac{23254349}{103680000}z^{16}w^{10}t^{2}+\frac{19269653381}{414720000}z^{15}w^{13}+\frac{355200857}{207360000}z^{15}w^{11}t^{2}+\frac{429876201377}{3317760000}z^{14}w^{14}+\frac{1800541141}{207360000}z^{14}w^{12}t^{2}+\frac{76870144463}{276480000}z^{13}w^{15}+\frac{318043699}{10368000}z^{13}w^{13}t^{2}+\frac{517037635199}{1105920000}z^{12}w^{16}+\frac{1629060539}{20736000}z^{12}w^{14}t^{2}+\frac{103567194539}{165888000}z^{11}w^{17}+\frac{31173563839}{207360000}z^{11}w^{15}t^{2}+\frac{9235476323}{13824000}z^{10}w^{18}+\frac{15185988517}{69120000}z^{10}w^{16}t^{2}+\frac{23876618441}{41472000}z^{9}w^{19}+\frac{25800087377}{103680000}z^{9}w^{17}t^{2}+\frac{27647801033}{69120000}z^{8}w^{20}+\frac{11419379159}{51840000}z^{8}w^{18}t^{2}+\frac{26805331}{120000}z^{7}w^{21}+\frac{3960499873}{25920000}z^{7}w^{19}t^{2}+\frac{322377811}{3240000}z^{6}w^{22}+\frac{178778401}{2160000}z^{6}w^{20}t^{2}+\frac{14122253}{405000}z^{5}w^{23}+\frac{18712087}{540000}z^{5}w^{21}t^{2}+\frac{3808517}{405000}z^{4}w^{24}+\frac{4453507}{405000}z^{4}w^{22}t^{2}+\frac{95344}{50625}z^{3}w^{25}+\frac{51799}{20250}z^{3}w^{23}t^{2}+\frac{1484}{5625}z^{2}w^{26}+\frac{20834}{50625}z^{2}w^{24}t^{2}+\frac{1168}{50625}zw^{27}+\frac{2072}{50625}zw^{25}t^{2}+\frac{16}{16875}w^{28}+\frac{32}{16875}w^{26}t^{2}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -\frac{1}{240}z^{4}w^{2}t+\frac{11}{60}z^{3}w^{3}t+\frac{2}{5}z^{2}w^{4}t+\frac{17}{60}zw^{5}t-\frac{1}{15}zw^{3}t^{3}+\frac{1}{15}w^{6}t-\frac{1}{15}w^{4}t^{3}$ |
The following modular covers realize this modular curve as a fiber product over $X(1)$.
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.
