$\GL_2(\Z/60\Z)$-generators: |
$\begin{bmatrix}29&35\\42&43\end{bmatrix}$, $\begin{bmatrix}43&16\\12&47\end{bmatrix}$, $\begin{bmatrix}47&20\\27&41\end{bmatrix}$, $\begin{bmatrix}49&38\\3&19\end{bmatrix}$ |
Contains $-I$: |
yes |
Quadratic refinements: |
60.64.1-60.a.1.1, 60.64.1-60.a.1.2, 60.64.1-60.a.1.3, 60.64.1-60.a.1.4, 60.64.1-60.a.1.5, 60.64.1-60.a.1.6, 60.64.1-60.a.1.7, 60.64.1-60.a.1.8, 120.64.1-60.a.1.1, 120.64.1-60.a.1.2, 120.64.1-60.a.1.3, 120.64.1-60.a.1.4, 120.64.1-60.a.1.5, 120.64.1-60.a.1.6, 120.64.1-60.a.1.7, 120.64.1-60.a.1.8 |
Cyclic 60-isogeny field degree: |
$36$ |
Cyclic 60-torsion field degree: |
$576$ |
Full 60-torsion field degree: |
$69120$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
$ 0 $ | $=$ | $ 3 x^{2} - 5 x y + x z - x w - 2 z w $ |
| $=$ | $10 x^{2} + 5 x y + 9 x z - x w + 5 y^{2} + 9 z^{2} + w^{2}$ |
Singular plane model Singular plane model
$ 0 $ | $=$ | $ 37 x^{4} + 28 x^{3} y - 8 x^{3} z + 23 x^{2} y^{2} - 12 x^{2} y z + 3 x^{2} z^{2} - 2 x y^{2} z + \cdots + 2 y^{2} z^{2} $ |
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:
$\displaystyle X$ |
$=$ |
$\displaystyle x$ |
$\displaystyle Y$ |
$=$ |
$\displaystyle z$ |
$\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Maps to other modular curves
$j$-invariant map
of degree 32 from the embedded model of this modular curve to the modular curve
$X(1)$
:
$\displaystyle j$ |
$=$ |
$\displaystyle \frac{2^4\cdot3^2}{5^2}\cdot\frac{142358832998694xz^{7}+47117090601180xz^{6}w+110216853019476xz^{5}w^{2}-27063942892230xz^{4}w^{3}-73404822589870xz^{3}w^{4}-19966819802196xz^{2}w^{5}-2410427149860xzw^{6}-328580678314xw^{7}+122628507375465y^{2}z^{6}-36366507296820y^{2}z^{5}w-73990382765625y^{2}z^{4}w^{2}-22224419438000y^{2}z^{3}w^{3}-21688048817925y^{2}z^{2}w^{4}-9098331659460y^{2}zw^{5}-813194336195y^{2}w^{6}+24457448136420yz^{6}w-21689189437500yz^{5}w^{2}-57544895809700yz^{4}w^{3}-13903814148900yz^{3}w^{4}+4382558458200yz^{2}w^{5}+1089764021320yzw^{6}+223433023184514z^{8}-107218106538354z^{7}w-120042222470154z^{6}w^{2}-37789085547306z^{5}w^{3}-50999509938240z^{4}w^{4}-12459252951406z^{3}w^{5}+1829322618846z^{2}w^{6}+689178517146zw^{7}+137551122614w^{8}}{437616168750xz^{7}-684500017500xz^{6}w+353677792500xz^{5}w^{2}-146476248750xz^{4}w^{3}+39100811994xz^{3}w^{4}-5064015444xz^{2}w^{5}-937510220xzw^{6}-106398482xw^{7}+331426434375y^{2}z^{6}-303556612500y^{2}z^{5}w+143965603125y^{2}z^{4}w^{2}-83997000000y^{2}z^{3}w^{3}+16988582025y^{2}z^{2}w^{4}-4964733220y^{2}zw^{5}+596476355y^{2}w^{6}+48964500000yz^{6}w+18269550000yz^{5}w^{2}+10780087500yz^{4}w^{3}-4605824340yz^{3}w^{4}+4868159660yz^{2}w^{5}-220320100yzw^{6}+596567581875z^{8}-669179576250z^{7}w+389619416250z^{6}w^{2}-218782316250z^{5}w^{3}+78250665078z^{4}w^{4}-29096940222z^{3}w^{5}+4640686938z^{2}w^{6}-1187473166zw^{7}+119295271w^{8}}$ |
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.