Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2-65392187626x-6436336642975927\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z-65392187626xz^2-6436336642975927z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-84748275163323x-300292451190557391978\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{585043893873306972750837829430365957644031405739624730621879266317137578803145397623779384121313615637458197731554220122530892085802191360983943411255796116471051}{603792022726368749033312963995366185242884835943432937769149138302706204026322343081709033836319840212143611887763764091484490789953580159185900179209235556}, \frac{429752827326037768132466391643214018393763668136759825906853159401073162935991567425348485069765544801113055335000884149961513024585755184014642817266075500636305182539219991540416439150544798156089626616046798156544732501514757562271847370119}{469170887904472226050980174507736847264660062192779555685828720519015992037649681149204504374623594282812253878124252406808552441215813301170098172056954877239145574391768760129701573998150286361320369608993721365165788515879261366296}\right)\) |
$\hat{h}(P)$ | ≈ | $369.40566243754808698712174170$ |
Torsion generators
\( \left(-\frac{590541}{4}, \frac{590537}{8}\right) \)
Integral points
None
Invariants
Conductor: | \( 66270 \) | = | $2 \cdot 3 \cdot 5 \cdot 47^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $51627761958004131705468750 $ | = | $2 \cdot 3^{10} \cdot 5^{8} \cdot 47^{9} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{27632526176252046076847}{46132031250} \) | = | $2^{-1} \cdot 3^{-10} \cdot 5^{-8} \cdot 3673^{3} \cdot 8231^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $4.6246706699720642757206800104\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $1.7370599686895203356049670081\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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||
$abc$ quality: | $1.0679091339854012\dots$ | |||
Szpiro ratio: | $7.775946186530531\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $369.40566243754808698712174170\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.0094410019667549474816192758949\dots$ | comment: Real Period
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
|
Tamagawa product: | $ 8 $ = $ 1\cdot2\cdot2\cdot2 $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
|
Torsion order: | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
Special value: | $ L'(E,1) $ ≈ $ 6.9751191712066114320361653007 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 6.975119171 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.009441 \cdot 369.405662 \cdot 8}{2^2} \approx 6.975119171$
Modular invariants
Modular form 66270.2.a.n
For more coefficients, see the Downloads section to the right.
Modular degree: | 181923840 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$3$ | $2$ | $I_{10}$ | Non-split multiplicative | 1 | 1 | 10 | 10 |
$5$ | $2$ | $I_{8}$ | Non-split multiplicative | 1 | 1 | 8 | 8 |
$47$ | $2$ | $III^{*}$ | Additive | -1 | 2 | 9 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
---|---|---|
$2$ | 2B | 8.6.0.3 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 376 = 2^{3} \cdot 47 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 236 & 145 \\ 329 & 48 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 20 & 1 \\ 183 & 0 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 187 & 0 \end{array}\right),\left(\begin{array}{rr} 373 & 4 \\ 372 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[376])$ is a degree-$611033088$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/376\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 66270p
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 66270v2, its twist by $-47$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{94}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$4$ | 4.0.3322336.1 | \(\Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.706426655801344.4 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | nonsplit | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord | ss | ord | add |
$\lambda$-invariant(s) | 8 | 1 | 1 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 | 1 | - |
$\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | - |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.