Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-569832x-165422656\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-569832xz^2-165422656z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-9117315x-10596167298\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(13691, 1592586\right)\) |
$\hat{h}(P)$ | ≈ | $6.4203188927015307207701057201$ |
Integral points
\( \left(13691, 1592586\right) \), \( \left(13691, -1606277\right) \)
Invariants
Conductor: | \( 85698 \) | = | $2 \cdot 3^{4} \cdot 23^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-13813524874368 $ | = | $-1 \cdot 2^{7} \cdot 3^{6} \cdot 23^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{189613868625}{128} \) | = | $-1 \cdot 2^{-7} \cdot 3^{3} \cdot 5^{3} \cdot 383^{3}$ | 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: | $1.8371151868422150272342166666\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.27993806545641466386678236777\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.1259568215438134\dots$ | |||
Szpiro ratio: | $4.522825406363255\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $6.4203188927015307207701057201\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.086882597649458473979514919320\dots$ | 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: | $ 6 $ = $ 1\cdot3\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)
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Torsion order: | $1$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 3.3468838988148230724694425228 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 3.346883899 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.086883 \cdot 6.420319 \cdot 6}{1^2} \approx 3.346883899$
Modular invariants
Modular form 85698.2.a.e
For more coefficients, see the Downloads section to the right.
Modular degree: | 498960 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{7}$ | Non-split multiplicative | 1 | 1 | 7 | 7 |
$3$ | $3$ | $IV$ | Additive | -1 | 4 | 6 | 0 |
$23$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 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 |
---|---|---|
$2$ | 2G | 8.2.0.1 |
$3$ | 3B | 3.4.0.1 |
$7$ | 7B | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 11592 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 23 \), index $768$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 3450 \\ 10626 & 5797 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 5544 & 1 \end{array}\right),\left(\begin{array}{rr} 9661 & 10626 \\ 7245 & 10627 \end{array}\right),\left(\begin{array}{rr} 1289 & 6440 \\ 5152 & 6441 \end{array}\right),\left(\begin{array}{rr} 11110 & 5865 \\ 7245 & 7729 \end{array}\right),\left(\begin{array}{rr} 6049 & 5544 \\ 6048 & 6049 \end{array}\right),\left(\begin{array}{rr} 6625 & 8970 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1007 & 0 \\ 0 & 11591 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 966 & 1 \end{array}\right),\left(\begin{array}{rr} 1473 & 7498 \\ 9016 & 4233 \end{array}\right),\left(\begin{array}{rr} 4831 & 10626 \\ 6279 & 6763 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1932 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 7728 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 967 & 1794 \\ 1932 & 4831 \end{array}\right),\left(\begin{array}{rr} 1 & 8280 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 7728 & 1 \end{array}\right),\left(\begin{array}{rr} 3865 & 7728 \\ 3864 & 7729 \end{array}\right)$.
The torsion field $K:=\Q(E[11592])$ is a degree-$4188236709888$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/11592\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 7 and 21.
Its isogeny class 85698j
consists of 4 curves linked by isogenies of
degrees dividing 21.
Twists
The minimal quadratic twist of this elliptic curve is 162c3, its twist by $-23$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-23}) \) | \(\Z/3\Z\) | Not in database |
$3$ | 3.1.648.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.3359232.4 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.2.425747664.3 | \(\Z/3\Z\) | Not in database |
$6$ | 6.0.4024991806227.4 | \(\Z/7\Z\) | Not in database |
$6$ | 6.0.5108971968.6 | \(\Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/21\Z\) | Not in database |
$18$ | 18.0.50452557693136526518161517430468352.2 | \(\Z/9\Z\) | Not in database |
$18$ | 18.2.74660227290948091740627460952162304.1 | \(\Z/6\Z\) | Not in database |
$18$ | 18.0.17093654581891449594915499403756652221693952.1 | \(\Z/14\Z\) | 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 | nonsplit | add | ss | ord | ord | ord | ord | ord | add | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 8 | - | 1,1 | 1 | 1 | 1 | 1 | 1 | - | 1 | 3 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | 0 | - | 0,0 | 1 | 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.