Embedded model Embedded model in $\mathbb{P}^{5}$
| $ 0 $ | $=$ | $ x^{2} + y z - y w - y t $ |
| $=$ | $x^{2} - x u + y^{2} - y z - y w$ |
| $=$ | $x y - 2 x z + x t + z u - w u - t u$ |
| $=$ | $x^{2} - x u - 3 y^{2} + u^{2}$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 225 x^{8} - 90 x^{7} y + 69 x^{6} y^{2} - 18 x^{6} z^{2} - 12 x^{5} y^{3} + 75 x^{5} y z^{2} + 4 x^{4} y^{4} + \cdots + z^{8} $ |
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Weierstrass model Weierstrass model
| $ y^{2} + \left(x^{4} + 1\right) y $ | $=$ | $ -2x^{8} + 10x^{7} - 34x^{6} + 70x^{5} - 93x^{4} + 70x^{3} - 34x^{2} + 10x - 2 $ |
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This modular curve has no real points, and therefore no rational points.
Maps to other modular curves
$j$-invariant map
of degree 72 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^2}{3}\cdot\frac{51918509760xwt^{6}u+124007565600xwt^{4}u^{3}+66843558720xwt^{2}u^{5}+9035569440xwu^{7}+13686202368xt^{7}u-53816228640xt^{5}u^{3}-66445770240xt^{3}u^{5}+31778341344xtu^{7}-20565110783yt^{8}+30700364455yt^{6}u^{2}+112231697895yt^{4}u^{4}+23052197421yt^{2}u^{6}-15409156320yu^{8}+9892438857zwt^{7}+6706026495zwt^{5}u^{2}-16529885505zwt^{3}u^{4}-8767426059zwtu^{6}+1477622286zt^{8}-28298574910zt^{6}u^{2}-72313215390zt^{4}u^{4}-9010033002zt^{2}u^{6}+4959780480zu^{8}-3790020381w^{2}t^{7}-19408309515w^{2}t^{5}u^{2}-34747489035w^{2}t^{3}u^{4}-28839086073w^{2}tu^{6}-3790020381wt^{8}-14841799195wt^{6}u^{2}+39479323365wt^{4}u^{4}+37925247447wt^{2}u^{6}-18988501680wu^{8}+10384979619t^{9}+41701506755t^{7}u^{2}+76726934775t^{5}u^{4}+26791250217t^{3}u^{6}-10670541930tu^{8}}{989280xwt^{6}u+29803800xwt^{4}u^{3}+202476960xwt^{2}u^{5}+58293720xwu^{7}+4314624xt^{7}u+101200280xt^{5}u^{3}+252668080xt^{3}u^{5}-73917528xtu^{7}-439194yt^{8}-31487010yt^{6}u^{2}-125722165yt^{4}u^{4}+12635448yt^{2}u^{6}+65469840yu^{8}-37674zwt^{7}-5722290zwt^{5}u^{2}-84302365zwt^{3}u^{4}-63120792zwtu^{6}+75348zt^{8}+3525420zt^{6}u^{2}+38789530zt^{4}u^{4}+22628424zt^{2}u^{6}-43331760zu^{8}-12558w^{2}t^{7}+170730w^{2}t^{5}u^{2}-2449055w^{2}t^{3}u^{4}-66535224w^{2}tu^{6}-12558wt^{8}-3641910wt^{6}u^{2}-71668855wt^{4}u^{4}-184341864wt^{2}u^{6}+61901160wu^{8}-12558t^{9}-2778210t^{7}u^{2}-49064275t^{5}u^{4}-59600254t^{3}u^{6}+75294360tu^{8}}$ |
Map
of degree 1 from the embedded model of this modular curve to the plane model of the modular curve
24.72.3.oe.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle u$ |
Equation of the image curve:
| $0$ |
$=$ |
$ 225X^{8}-90X^{7}Y+69X^{6}Y^{2}-12X^{5}Y^{3}+4X^{4}Y^{4}-18X^{6}Z^{2}+75X^{5}YZ^{2}-12X^{4}Y^{2}Z^{2}-2X^{3}Y^{3}Z^{2}+6X^{4}Z^{4}-24X^{3}YZ^{4}+3X^{2}Y^{2}Z^{4}-6X^{2}Z^{6}+XYZ^{6}+Z^{8} $ |
Map
of degree 1 from the embedded model of this modular curve to the Weierstrass model of the modular curve
24.72.3.oe.1
:
| $\displaystyle X$ |
$=$ |
$\displaystyle \frac{5}{34}y^{6}+\frac{9}{34}y^{5}t-\frac{1}{2}y^{5}u-\frac{2}{51}y^{4}t^{2}+\frac{25}{34}y^{4}u^{2}+\frac{2}{51}y^{3}t^{3}-\frac{10}{51}y^{3}tu^{2}-\frac{1}{17}y^{3}u^{3}-\frac{1}{51}y^{2}t^{2}u^{2}-\frac{13}{102}y^{2}u^{4}+\frac{5}{102}ytu^{4}+\frac{1}{34}yu^{5}-\frac{1}{102}u^{6}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle -\frac{125}{39304}y^{23}u-\frac{675}{39304}y^{22}tu+\frac{3575}{78608}y^{22}u^{2}-\frac{1115}{39304}y^{21}t^{2}u+\frac{3105}{19652}y^{21}tu^{2}-\frac{4459605}{668168}y^{21}u^{3}-\frac{469}{39304}y^{20}t^{3}u+\frac{8123}{78608}y^{20}t^{2}u^{2}+\frac{1011743}{334084}y^{20}tu^{3}-\frac{1764255}{668168}y^{20}u^{4}-\frac{47}{29478}y^{19}t^{4}u-\frac{1285}{39304}y^{19}t^{3}u^{2}-\frac{2256011}{2004504}y^{19}t^{2}u^{3}-\frac{3550957}{1336336}y^{19}tu^{4}+\frac{221287}{334084}y^{19}u^{5}-\frac{239}{29478}y^{18}t^{5}u+\frac{716}{14739}y^{18}t^{4}u^{2}+\frac{246521}{1002252}y^{18}t^{3}u^{3}-\frac{634803}{1336336}y^{18}t^{2}u^{4}-\frac{794389}{2004504}y^{18}tu^{5}-\frac{1864365}{1336336}y^{18}u^{6}+\frac{242}{132651}y^{17}t^{6}u-\frac{443}{29478}y^{17}t^{5}u^{2}+\frac{15571}{1002252}y^{17}t^{4}u^{3}+\frac{25453}{1002252}y^{17}t^{3}u^{4}-\frac{176903}{250563}y^{17}t^{2}u^{5}-\frac{1416603}{1336336}y^{17}tu^{6}+\frac{2316397}{2004504}y^{17}u^{7}-\frac{62}{44217}y^{16}t^{7}u+\frac{701}{132651}y^{16}t^{6}u^{2}+\frac{6899}{751689}y^{16}t^{5}u^{3}-\frac{54095}{1002252}y^{16}t^{4}u^{4}+\frac{2252}{83521}y^{16}t^{3}u^{5}+\frac{404353}{2004504}y^{16}t^{2}u^{6}-\frac{147889}{167042}y^{16}tu^{7}+\frac{146673}{334084}y^{16}u^{8}+\frac{8}{44217}y^{15}t^{8}u-\frac{62}{44217}y^{15}t^{7}u^{2}+\frac{2116}{2255067}y^{15}t^{6}u^{3}+\frac{8276}{751689}y^{15}t^{5}u^{4}+\frac{10453}{1503378}y^{15}t^{4}u^{5}-\frac{27095}{250563}y^{15}t^{3}u^{6}+\frac{60173}{501126}y^{15}t^{2}u^{7}+\frac{1793341}{2004504}y^{15}tu^{8}-\frac{163431}{334084}y^{15}u^{9}-\frac{8}{132651}y^{14}t^{9}u+\frac{8}{44217}y^{14}t^{8}u^{2}+\frac{140}{250563}y^{14}t^{7}u^{3}-\frac{3851}{2255067}y^{14}t^{6}u^{4}-\frac{6778}{751689}y^{14}t^{5}u^{5}+\frac{69965}{3006756}y^{14}t^{4}u^{6}-\frac{187651}{9020268}y^{14}t^{3}u^{7}-\frac{87833}{2004504}y^{14}t^{2}u^{8}+\frac{192901}{1002252}y^{14}tu^{9}+\frac{256093}{668168}y^{14}u^{10}-\frac{8}{132651}y^{13}t^{9}u^{2}+\frac{28}{250563}y^{13}t^{8}u^{3}+\frac{140}{250563}y^{13}t^{7}u^{4}+\frac{1582}{2255067}y^{13}t^{6}u^{5}-\frac{3025}{751689}y^{13}t^{5}u^{6}-\frac{8927}{1002252}y^{13}t^{4}u^{7}+\frac{223451}{9020268}y^{13}t^{3}u^{8}+\frac{484247}{3006756}y^{13}t^{2}u^{9}-\frac{3301}{668168}y^{13}tu^{10}-\frac{7609}{334084}y^{13}u^{11}-\frac{16}{2255067}y^{12}t^{9}u^{3}+\frac{28}{250563}y^{12}t^{8}u^{4}-\frac{26}{250563}y^{12}t^{7}u^{5}-\frac{1064}{2255067}y^{12}t^{6}u^{6}-\frac{203}{751689}y^{12}t^{5}u^{7}-\frac{278}{250563}y^{12}t^{4}u^{8}-\frac{131023}{9020268}y^{12}t^{3}u^{9}+\frac{64751}{3006756}y^{12}t^{2}u^{10}+\frac{78284}{751689}y^{12}tu^{11}-\frac{43985}{501126}y^{12}u^{12}-\frac{16}{2255067}y^{11}t^{9}u^{4}-\frac{26}{250563}y^{11}t^{7}u^{6}+\frac{113}{2255067}y^{11}t^{6}u^{7}+\frac{211}{751689}y^{11}t^{5}u^{8}+\frac{4501}{1503378}y^{11}t^{4}u^{9}+\frac{62855}{9020268}y^{11}t^{3}u^{10}-\frac{70945}{3006756}y^{11}t^{2}u^{11}-\frac{97481}{1503378}y^{11}tu^{12}+\frac{156772}{2255067}y^{11}u^{13}+\frac{8}{2255067}y^{10}t^{9}u^{5}-\frac{56}{751689}y^{10}t^{7}u^{7}+\frac{815}{2255067}y^{10}t^{6}u^{8}+\frac{1055}{1503378}y^{10}t^{5}u^{9}-\frac{3257}{1503378}y^{10}t^{4}u^{10}+\frac{907}{530604}y^{10}t^{3}u^{11}+\frac{275}{3006756}y^{10}t^{2}u^{12}-\frac{116477}{3006756}y^{10}tu^{13}-\frac{449749}{18040536}y^{10}u^{14}+\frac{8}{2255067}y^{9}t^{9}u^{6}-\frac{4}{751689}y^{9}t^{8}u^{7}-\frac{56}{751689}y^{9}t^{7}u^{8}-\frac{94}{2255067}y^{9}t^{6}u^{9}+\frac{407}{1503378}y^{9}t^{5}u^{10}+\frac{1591}{3006756}y^{9}t^{4}u^{11}-\frac{22237}{9020268}y^{9}t^{3}u^{12}-\frac{2071}{250563}y^{9}t^{2}u^{13}+\frac{27097}{3006756}y^{9}tu^{14}-\frac{101291}{9020268}y^{9}u^{15}-\frac{4}{751689}y^{8}t^{8}u^{8}+\frac{4}{250563}y^{8}t^{7}u^{9}+\frac{41}{2255067}y^{8}t^{6}u^{10}-\frac{80}{751689}y^{8}t^{5}u^{11}+\frac{7}{10404}y^{8}t^{4}u^{12}+\frac{2389}{2255067}y^{8}t^{3}u^{13}-\frac{2239}{668168}y^{8}t^{2}u^{14}-\frac{1655}{1503378}y^{8}tu^{15}+\frac{76315}{9020268}y^{8}u^{16}+\frac{4}{250563}y^{7}t^{7}u^{10}-\frac{1}{2255067}y^{7}t^{6}u^{11}-\frac{107}{751689}y^{7}t^{5}u^{12}-\frac{155}{501126}y^{7}t^{4}u^{13}+\frac{499}{2255067}y^{7}t^{3}u^{14}+\frac{2945}{1503378}y^{7}t^{2}u^{15}+\frac{2113}{6013512}y^{7}tu^{16}-\frac{41551}{18040536}y^{7}u^{17}-\frac{28}{2255067}y^{6}t^{6}u^{12}+\frac{5}{751689}y^{6}t^{5}u^{13}-\frac{37}{1002252}y^{6}t^{4}u^{14}-\frac{1721}{9020268}y^{6}t^{3}u^{15}+\frac{4919}{6013512}y^{6}t^{2}u^{16}+\frac{12845}{6013512}y^{6}tu^{17}-\frac{9563}{36081072}y^{6}u^{18}+\frac{14}{751689}y^{5}t^{5}u^{14}+\frac{5}{176868}y^{5}t^{4}u^{15}-\frac{299}{9020268}y^{5}t^{3}u^{16}-\frac{667}{6013512}y^{5}t^{2}u^{17}-\frac{157}{6013512}y^{5}tu^{18}+\frac{13709}{18040536}y^{5}u^{19}-\frac{7}{1503378}y^{4}t^{4}u^{16}+\frac{125}{18040536}y^{4}t^{3}u^{17}-\frac{647}{12027024}y^{4}t^{2}u^{18}-\frac{967}{3006756}y^{4}tu^{19}-\frac{3175}{18040536}y^{4}u^{20}+\frac{89}{18040536}y^{3}t^{3}u^{18}+\frac{1}{2004504}y^{3}t^{2}u^{19}-\frac{307}{12027024}y^{3}tu^{20}-\frac{545}{9020268}y^{3}u^{21}-\frac{1}{1336336}y^{2}t^{2}u^{20}+\frac{89}{6013512}y^{2}tu^{21}+\frac{1033}{36081072}y^{2}u^{22}+\frac{31}{12027024}ytu^{22}+\frac{1}{1061208}yu^{23}-\frac{7}{4510134}u^{24}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{2}y^{5}u+\frac{1}{2}y^{4}u^{2}+\frac{1}{17}y^{3}u^{3}+\frac{1}{17}y^{2}u^{4}-\frac{1}{34}yu^{5}-\frac{1}{34}u^{6}$ |
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.
