Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2+9555583x+26802839441\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z+9555583xz^2+26802839441z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+152889333x+1715534613574\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{9}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(4681, 414894\right)\) |
$\hat{h}(P)$ | ≈ | $2.1449540424403794491872357771$ |
Torsion generators
\( \left(-1689, 77284\right) \)
Integral points
\( \left(-1689, 77284\right) \), \( \left(-1689, -75596\right) \), \( \left(-1479, 97864\right) \), \( \left(-1479, -96386\right) \), \( \left(-129, 159964\right) \), \( \left(-129, -159836\right) \), \( \left(271, 171364\right) \), \( \left(271, -171636\right) \), \( \left(3015, 286612\right) \), \( \left(3015, -289628\right) \), \( \left(3771, 339364\right) \), \( \left(3771, -343136\right) \), \( \left(4681, 414894\right) \), \( \left(4681, -419576\right) \), \( \left(12871, 1504164\right) \), \( \left(12871, -1517036\right) \), \( \left(17421, 2332264\right) \), \( \left(17421, -2349686\right) \), \( \left(24771, 3919864\right) \), \( \left(24771, -3944636\right) \), \( \left(686271, 568179364\right) \), \( \left(686271, -568865636\right) \)
Invariants
Conductor: | \( 417690 \) | = | $2 \cdot 3^{3} \cdot 5 \cdot 7 \cdot 13 \cdot 17$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-366241548885849000000000 $ | = | $-1 \cdot 2^{9} \cdot 3^{5} \cdot 5^{9} \cdot 7^{9} \cdot 13^{3} \cdot 17 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{397090089854833871209893}{1507166867843000000000} \) | = | $2^{-9} \cdot 3 \cdot 5^{-9} \cdot 7^{-9} \cdot 13^{-3} \cdot 17^{-1} \cdot 19^{3} \cdot 2682269^{3}$ | 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);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $3.2042656747708444008391254278\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.7465105544924653627577732457\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9848336046012909\dots$ | |||
Szpiro ratio: | $4.755410145297052\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.1449540424403794491872357771\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.067952318609954455466283152536\dots$ | 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);
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Tamagawa product: | $ 6561 $ = $ 3^{2}\cdot3\cdot3^{2}\cdot3^{2}\cdot3\cdot1 $ | 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: | $9$ | 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) $ ≈ $ 11.806122640145093261629587292 $ | 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
$\displaystyle 11.806122640 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.067952 \cdot 2.144954 \cdot 6561}{9^2} \approx 11.806122640$
Modular invariants
Modular form 417690.2.a.cy
For more coefficients, see the Downloads section to the right.
Modular degree: | 85660416 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | not computed* (one of 2 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 6 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $9$ | $I_{9}$ | Split multiplicative | -1 | 1 | 9 | 9 |
$3$ | $3$ | $IV$ | Additive | 1 | 3 | 5 | 0 |
$5$ | $9$ | $I_{9}$ | Split multiplicative | -1 | 1 | 9 | 9 |
$7$ | $9$ | $I_{9}$ | Split multiplicative | -1 | 1 | 9 | 9 |
$13$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$17$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
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 |
---|---|---|
$3$ | 3B.1.1 | 9.72.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 556920 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \cdot 17 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 556903 & 18 \\ 556902 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 477361 & 18 \\ 397809 & 163 \end{array}\right),\left(\begin{array}{rr} 139231 & 18 \\ 139239 & 163 \end{array}\right),\left(\begin{array}{rr} 445537 & 18 \\ 111393 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 65521 & 18 \\ 32769 & 163 \end{array}\right),\left(\begin{array}{rr} 278461 & 18 \\ 278469 & 163 \end{array}\right),\left(\begin{array}{rr} 139241 & 278478 \\ 418023 & 480115 \end{array}\right),\left(\begin{array}{rr} 342721 & 18 \\ 299889 & 163 \end{array}\right)$.
The torsion field $K:=\Q(E[556920])$ is a degree-$82391425496886804480$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/556920\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 417690.cy
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 417690.v3, its twist by $-3$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.