$\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}1&5\\48&23\end{bmatrix}$, $\begin{bmatrix}7&0\\32&31\end{bmatrix}$, $\begin{bmatrix}7&30\\30&23\end{bmatrix}$, $\begin{bmatrix}39&20\\16&39\end{bmatrix}$, $\begin{bmatrix}51&35\\4&29\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
60.144.1-60.d.1.1, 60.144.1-60.d.1.2, 60.144.1-60.d.1.3, 60.144.1-60.d.1.4, 60.144.1-60.d.1.5, 60.144.1-60.d.1.6, 60.144.1-60.d.1.7, 60.144.1-60.d.1.8, 120.144.1-60.d.1.1, 120.144.1-60.d.1.2, 120.144.1-60.d.1.3, 120.144.1-60.d.1.4, 120.144.1-60.d.1.5, 120.144.1-60.d.1.6, 120.144.1-60.d.1.7, 120.144.1-60.d.1.8 |
Cyclic 60-isogeny field degree: |
$8$ |
Cyclic 60-torsion field degree: |
$128$ |
Full 60-torsion field degree: |
$30720$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 4 y^{2} - y z + z^{2} + w^{2} $ |
| $=$ | $15 x^{2} - y w + 2 z w - w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 20 x^{4} + 15 x^{2} y z - 24 x^{2} z^{2} + 3 y^{2} z^{2} - 9 y z^{3} + 9 z^{4} $ |
This modular curve has no real points, and therefore no rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle \frac{1}{3}w$ |
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{3^3}{2^2}\cdot\frac{190153669921875yz^{17}-64435585556250000yz^{16}w+6908354275387500000yz^{15}w^{2}-243599175478125000000yz^{14}w^{3}+1330470514205268750000yz^{13}w^{4}-2443747148801532000000yz^{12}w^{5}+2571149604507084000000yz^{11}w^{6}-5483665617448147200000yz^{10}w^{7}+10627118767735524000000yz^{9}w^{8}-16102738315610918400000yz^{8}w^{9}+16179546468938757120000yz^{7}w^{10}-11650559372883267584000yz^{6}w^{11}+5501881036696200192000yz^{5}w^{12}-1574586681932660736000yz^{4}w^{13}+135628810672717824000yz^{3}w^{14}+83985339416696586240yz^{2}w^{15}-7155095468769607680yzw^{16}-149134468753391616yw^{17}-39017162109375z^{18}+21866523581250000z^{17}w-3510300720779296875z^{16}w^{2}+185927805026831250000z^{15}w^{3}-1688650306617473437500z^{14}w^{4}+6296639441760234000000z^{13}w^{5}-13218138029403110250000z^{12}w^{6}+20743160012200382400000z^{11}w^{7}-26004260392040077200000z^{10}w^{8}+24416424925383436800000z^{9}w^{9}-18637659240494729760000z^{8}w^{10}+9275576205802943488000z^{7}w^{11}-3218193944186100480000z^{6}w^{12}-186792662153748480000z^{5}w^{13}+491356820073636864000z^{4}w^{14}-291383597509612666880z^{3}w^{15}+32520211290972487680z^{2}w^{16}+2908146245484675072zw^{17}-134467123384877056w^{18}}{w(614322319921875yz^{16}+9521599992187500yz^{15}w+57977447228437500yz^{14}w^{2}+175059645187500000yz^{13}w^{3}+294326568803250000yz^{12}w^{4}+497455470522000000yz^{11}w^{5}+1448675929911600000yz^{10}w^{6}+2764626408144000000yz^{9}w^{7}+1311026976727200000yz^{8}w^{8}-2982185665056000000yz^{7}w^{9}-3565212598861056000yz^{6}w^{10}+422161442887680000yz^{5}w^{11}+1696805664692428800yz^{4}w^{12}+64794880696320000yz^{3}w^{13}-266703241855303680yz^{2}w^{14}+13147084176752640yzw^{15}+3738339953278976yw^{16}-208473212109375z^{17}-4838481210937500z^{16}w-43303298643984375z^{15}w^{2}-199028157117187500z^{14}w^{3}-508270927188375000z^{13}w^{4}-700312839788250000z^{12}w^{5}-400831745483700000z^{11}w^{6}+217495217566800000z^{10}w^{7}+950410250085600000z^{9}w^{8}+1813109124060000000z^{8}w^{9}+1370877604656192000z^{7}w^{10}-829334545862400000z^{6}w^{11}-1478161087719321600z^{5}w^{12}-12443261230080000z^{4}w^{13}+417477542203637760z^{3}w^{14}-4718559305072640z^{2}w^{15}-25492207040069632zw^{16}+1011706805026816w^{17})}$ |
Hi
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Cover information
Click on a modular curve in the diagram to see information about it.
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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.