| $\GL_2(\Z/8\Z)$-generators: |
$\begin{bmatrix}1&0\\0&7\end{bmatrix}$, $\begin{bmatrix}1&0\\4&3\end{bmatrix}$, $\begin{bmatrix}1&4\\4&1\end{bmatrix}$, $\begin{bmatrix}7&4\\4&1\end{bmatrix}$ |
| $\GL_2(\Z/8\Z)$-subgroup: |
$C_2^4$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
8.192.3-8.j.1.1, 8.192.3-8.j.1.2, 8.192.3-8.j.1.3, 8.192.3-8.j.1.4, 8.192.3-8.j.1.5, 8.192.3-8.j.1.6, 16.192.3-8.j.1.1, 16.192.3-8.j.1.2, 16.192.3-8.j.1.3, 16.192.3-8.j.1.4, 24.192.3-8.j.1.1, 24.192.3-8.j.1.2, 24.192.3-8.j.1.3, 24.192.3-8.j.1.4, 24.192.3-8.j.1.5, 24.192.3-8.j.1.6, 40.192.3-8.j.1.1, 40.192.3-8.j.1.2, 40.192.3-8.j.1.3, 40.192.3-8.j.1.4, 40.192.3-8.j.1.5, 40.192.3-8.j.1.6, 48.192.3-8.j.1.1, 48.192.3-8.j.1.2, 48.192.3-8.j.1.3, 48.192.3-8.j.1.4, 56.192.3-8.j.1.1, 56.192.3-8.j.1.2, 56.192.3-8.j.1.3, 56.192.3-8.j.1.4, 56.192.3-8.j.1.5, 56.192.3-8.j.1.6, 80.192.3-8.j.1.1, 80.192.3-8.j.1.2, 80.192.3-8.j.1.3, 80.192.3-8.j.1.4, 88.192.3-8.j.1.1, 88.192.3-8.j.1.2, 88.192.3-8.j.1.3, 88.192.3-8.j.1.4, 88.192.3-8.j.1.5, 88.192.3-8.j.1.6, 104.192.3-8.j.1.1, 104.192.3-8.j.1.2, 104.192.3-8.j.1.3, 104.192.3-8.j.1.4, 104.192.3-8.j.1.5, 104.192.3-8.j.1.6, 112.192.3-8.j.1.1, 112.192.3-8.j.1.2, 112.192.3-8.j.1.3, 112.192.3-8.j.1.4, 120.192.3-8.j.1.1, 120.192.3-8.j.1.2, 120.192.3-8.j.1.3, 120.192.3-8.j.1.4, 120.192.3-8.j.1.5, 120.192.3-8.j.1.6, 136.192.3-8.j.1.1, 136.192.3-8.j.1.2, 136.192.3-8.j.1.3, 136.192.3-8.j.1.4, 136.192.3-8.j.1.5, 136.192.3-8.j.1.6, 152.192.3-8.j.1.1, 152.192.3-8.j.1.2, 152.192.3-8.j.1.3, 152.192.3-8.j.1.4, 152.192.3-8.j.1.5, 152.192.3-8.j.1.6, 168.192.3-8.j.1.1, 168.192.3-8.j.1.2, 168.192.3-8.j.1.3, 168.192.3-8.j.1.4, 168.192.3-8.j.1.5, 168.192.3-8.j.1.6, 176.192.3-8.j.1.1, 176.192.3-8.j.1.2, 176.192.3-8.j.1.3, 176.192.3-8.j.1.4, 184.192.3-8.j.1.1, 184.192.3-8.j.1.2, 184.192.3-8.j.1.3, 184.192.3-8.j.1.4, 184.192.3-8.j.1.5, 184.192.3-8.j.1.6, 208.192.3-8.j.1.1, 208.192.3-8.j.1.2, 208.192.3-8.j.1.3, 208.192.3-8.j.1.4, 232.192.3-8.j.1.1, 232.192.3-8.j.1.2, 232.192.3-8.j.1.3, 232.192.3-8.j.1.4, 232.192.3-8.j.1.5, 232.192.3-8.j.1.6, 240.192.3-8.j.1.1, 240.192.3-8.j.1.2, 240.192.3-8.j.1.3, 240.192.3-8.j.1.4, 248.192.3-8.j.1.1, 248.192.3-8.j.1.2, 248.192.3-8.j.1.3, 248.192.3-8.j.1.4, 248.192.3-8.j.1.5, 248.192.3-8.j.1.6, 264.192.3-8.j.1.1, 264.192.3-8.j.1.2, 264.192.3-8.j.1.3, 264.192.3-8.j.1.4, 264.192.3-8.j.1.5, 264.192.3-8.j.1.6, 272.192.3-8.j.1.1, 272.192.3-8.j.1.2, 272.192.3-8.j.1.3, 272.192.3-8.j.1.4, 280.192.3-8.j.1.1, 280.192.3-8.j.1.2, 280.192.3-8.j.1.3, 280.192.3-8.j.1.4, 280.192.3-8.j.1.5, 280.192.3-8.j.1.6, 296.192.3-8.j.1.1, 296.192.3-8.j.1.2, 296.192.3-8.j.1.3, 296.192.3-8.j.1.4, 296.192.3-8.j.1.5, 296.192.3-8.j.1.6, 304.192.3-8.j.1.1, 304.192.3-8.j.1.2, 304.192.3-8.j.1.3, 304.192.3-8.j.1.4, 312.192.3-8.j.1.1, 312.192.3-8.j.1.2, 312.192.3-8.j.1.3, 312.192.3-8.j.1.4, 312.192.3-8.j.1.5, 312.192.3-8.j.1.6, 328.192.3-8.j.1.1, 328.192.3-8.j.1.2, 328.192.3-8.j.1.3, 328.192.3-8.j.1.4, 328.192.3-8.j.1.5, 328.192.3-8.j.1.6 |
| Cyclic 8-isogeny field degree: |
$2$ |
| Cyclic 8-torsion field degree: |
$4$ |
| Full 8-torsion field degree: |
$16$ |
Embedded model Embedded model in $\mathbb{P}^{4}$
| $ 0 $ | $=$ | $ x^{2} y + x^{2} z - x y^{2} - x z^{2} - y^{2} z - z^{3} $ |
| $=$ | $x y^{2} - x y t + x z^{2} - x z t - y^{3} + y^{2} t - y z^{2} + y z t + z^{2} w + z^{2} t$ |
| $=$ | $2 x^{2} y + x y w - x y t - y^{3} + y^{2} t - y z^{2} + y z w$ |
| $=$ | $x^{2} y - x^{2} z - x y^{2} - x z^{2} - x z w + x z t - y z t - z^{2} w$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{6} + 6 x^{5} z - 4 x^{4} y^{2} - 16 x^{4} y z - x^{4} z^{2} + 4 x^{3} y^{3} + 8 x^{3} y^{2} z + \cdots - 11 z^{6} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ -2x^{7} + 14x^{5} - 14x^{3} + 2x $ |
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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 |
|---|
| $(1/2:1/2:-1/2:-1:1)$, $(0:0:0:-1:1)$, $(-1/2:0:0:1:0)$, $(1/2:0:0:0:1)$ |
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle t$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle z$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ |
$=$ |
$\displaystyle -\frac{1}{2}y^{5}-\frac{7}{2}y^{4}z-\frac{1}{2}y^{4}t-3y^{3}z^{2}+4y^{3}zt+\frac{3}{2}y^{3}t^{2}-11y^{2}z^{3}+2y^{2}z^{2}t+\frac{3}{2}y^{2}zt^{2}-\frac{1}{2}y^{2}t^{3}-\frac{9}{2}yz^{4}+6yz^{3}t+\frac{5}{2}yz^{2}t^{2}-yzt^{3}-\frac{19}{2}z^{5}+\frac{1}{2}z^{4}t+\frac{5}{2}z^{3}t^{2}-\frac{1}{2}z^{2}t^{3}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle 8y^{13}z^{7}+120y^{12}z^{8}+8y^{12}z^{7}t+752y^{11}z^{9}-24y^{11}z^{7}t^{2}+3024y^{10}z^{10}-272y^{10}z^{9}t-216y^{10}z^{8}t^{2}+8y^{10}z^{7}t^{3}+8872y^{9}z^{11}-1632y^{9}z^{10}t-1008y^{9}z^{9}t^{2}+80y^{9}z^{8}t^{3}+20472y^{8}z^{12}-5288y^{8}z^{11}t-3120y^{8}z^{10}t^{2}+392y^{8}z^{9}t^{3}+37728y^{7}z^{13}-12160y^{7}z^{12}t-7248y^{7}z^{11}t^{2}+1216y^{7}z^{10}t^{3}+56480y^{6}z^{14}-20832y^{6}z^{13}t-13008y^{6}z^{12}t^{2}+2704y^{6}z^{11}t^{3}+69400y^{5}z^{15}-26432y^{5}z^{14}t-17920y^{5}z^{13}t^{2}+4576y^{5}z^{12}t^{3}+68648y^{4}z^{16}-25544y^{4}z^{15}t-19328y^{4}z^{14}t^{2}+5840y^{4}z^{13}t^{3}+53808y^{3}z^{17}-18816y^{3}z^{16}t-16600y^{3}z^{15}t^{2}+5312y^{3}z^{14}t^{3}+32784y^{2}z^{18}-9360y^{2}z^{17}t-10648y^{2}z^{16}t^{2}+3176y^{2}z^{15}t^{3}+13752yz^{19}-2400yz^{18}t-4304yz^{17}t^{2}+1104yz^{16}t^{3}+2792z^{20}-152z^{19}t-784z^{18}t^{2}+168z^{17}t^{3}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle y^{3}z^{2}+3y^{2}z^{3}+3yz^{4}+z^{5}$ |
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}{2^3}\cdot\frac{4883415782144xzt^{12}+812714xw^{13}+293392xw^{12}t+14058370xw^{11}t^{2}+90026416xw^{10}t^{3}+216221656xw^{9}t^{4}-1212126208xw^{8}t^{5}-8384961124xw^{7}t^{6}-27019395768xw^{6}t^{7}-71007691902xw^{5}t^{8}-180146288440xw^{4}t^{9}-435565507614xw^{3}t^{10}-1023765901656xw^{2}t^{11}-1594636997508xwt^{12}+350879496648xt^{13}-4294967296yz^{13}+25769803776yz^{12}t+30064771072yz^{11}t^{2}-10737418240yz^{10}t^{3}-31943819264yz^{9}t^{4}-37044092928yz^{8}t^{5}-26575110144yz^{7}t^{6}+2013265920yz^{6}t^{7}+45222985728yz^{5}t^{8}+88415928320yz^{4}t^{9}+102416515072yz^{3}t^{10}+56845402112yz^{2}t^{11}-3421403010304yzt^{12}+208175yw^{13}+13745750yw^{12}t+132152225yw^{11}t^{2}+597841059yw^{10}t^{3}+1659187466yw^{9}t^{4}+3432398323yw^{8}t^{5}+7726691354yw^{7}t^{6}+21361034782yw^{6}t^{7}+55915783307yw^{5}t^{8}+131397459388yw^{4}t^{9}+304054659221yw^{3}t^{10}+719536265455yw^{2}t^{11}+1113399573516ywt^{12}-251056687701yt^{13}+8589934592z^{14}+17179869184z^{13}t-30064771072z^{12}t^{2}-36507222016z^{11}t^{3}-18522046464z^{10}t^{4}+9663676416z^{9}t^{5}+37849399296z^{8}t^{6}+53016002560z^{7}t^{7}+49450844160z^{6}t^{8}+14713618432z^{5}t^{9}-63963136000z^{4}t^{10}-155562541056z^{3}t^{11}+3238088z^{2}w^{12}+6490312z^{2}w^{11}t-387763520z^{2}w^{10}t^{2}-2834560104z^{2}w^{9}t^{3}-5554366904z^{2}w^{8}t^{4}-2762895568z^{2}w^{7}t^{5}-17779316832z^{2}w^{6}t^{6}-131788007216z^{2}w^{5}t^{7}-389007964552z^{2}w^{4}t^{8}-769114411768z^{2}w^{3}t^{9}-1627347185888z^{2}w^{2}t^{10}-3982614295656z^{2}wt^{11}-1460213718024z^{2}t^{12}-2952935zw^{13}-10119546zw^{12}t+167749899zw^{11}t^{2}+549341783zw^{10}t^{3}-1051127118zw^{9}t^{4}-7079368817zw^{8}t^{5}-196671578zw^{7}t^{6}+45659366238zw^{6}t^{7}+67374831981zw^{5}t^{8}-23695042452zw^{4}t^{9}-17434372193zw^{3}t^{10}+333661892443zw^{2}t^{11}-674340272360zwt^{12}-603988582385zt^{13}+414549w^{14}-1319171w^{13}t-5330053w^{12}t^{2}+71427599w^{11}t^{3}+334168710w^{10}t^{4}-516054598w^{9}t^{5}-6250424494w^{8}t^{6}-14790544250w^{7}t^{7}-14536705647w^{6}t^{8}-25901216535w^{5}t^{9}-123669502841w^{4}t^{10}-343711510373w^{3}t^{11}-312352039228w^{2}t^{12}-247069843504wt^{13}-175439772900t^{14}}{234499584xzt^{12}-126xw^{13}-2064xw^{12}t-12588xw^{11}t^{2}-41060xw^{10}t^{3}-100914xw^{9}t^{4}-258212xw^{8}t^{5}-645784xw^{7}t^{6}-1476568xw^{6}t^{7}-3442162xw^{5}t^{8}-8390632xw^{4}t^{9}-20313148xw^{3}t^{10}-48698436xw^{2}t^{11}-76053598xwt^{12}+17059356xt^{13}-131072yz^{5}t^{8}+786432yz^{4}t^{9}+1441792yz^{3}t^{10}+983040yz^{2}t^{11}-165233152yzt^{12}+45yw^{13}+1035yw^{12}t+8844yw^{11}t^{2}+39284yw^{10}t^{3}+106389yw^{9}t^{4}+210523yw^{8}t^{5}+415040yw^{7}t^{6}+989360yw^{6}t^{7}+2479159yw^{5}t^{8}+5988817yw^{4}t^{9}+14321716yw^{3}t^{10}+34404828yw^{2}t^{11}+53545223ywt^{12}-12107831yt^{13}+262144z^{6}t^{8}+524288z^{5}t^{9}-655360z^{4}t^{10}-2555904z^{3}t^{11}+192z^{2}w^{12}-3040z^{2}w^{11}t-63184z^{2}w^{10}t^{2}-364928z^{2}w^{9}t^{3}-939408z^{2}w^{8}t^{4}-1198592z^{2}w^{7}t^{5}-1551712z^{2}w^{6}t^{6}-5261760z^{2}w^{5}t^{7}-15168288z^{2}w^{4}t^{8}-34374432z^{2}w^{3}t^{9}-79646672z^{2}w^{2}t^{10}-193322944z^{2}wt^{11}-69192720z^{2}t^{12}-65zw^{13}+1807zw^{12}t+28608zw^{11}t^{2}+122344zw^{10}t^{3}+114579zw^{9}t^{4}-383709zw^{8}t^{5}-615896zw^{7}t^{6}+1018696zw^{6}t^{7}+2669109zw^{5}t^{8}+1594421zw^{4}t^{9}+3344280zw^{3}t^{10}+14824272zw^{2}t^{11}-34994183zwt^{12}-29221095zt^{13}-63w^{14}-1087w^{13}t-5970w^{12}t^{2}-9224w^{11}t^{3}+18277w^{10}t^{4}+31677w^{9}t^{5}-214530w^{8}t^{6}-749576w^{7}t^{7}-1113877w^{6}t^{8}-1914309w^{5}t^{9}-5712094w^{4}t^{10}-14994864w^{3}t^{11}-14120913w^{2}t^{12}-12041465wt^{13}-8529678t^{14}}$ |
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Cover information
Click on a modular curve in the diagram to see information about it.
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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.
