Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-4967137937x-135024131935319\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-4967137937xz^2-135024131935319z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6437410766379x-6299666587341944154\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(591670, 451427619\right)\)
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| $\hat{h}(P)$ | ≈ | $2.0680831199470976407897520607$ |
Integral points
\( \left(591670, 451427619\right) \), \( \left(591670, -452019289\right) \)
Invariants
| Conductor: | \( 440818 \) | = | $2 \cdot 7 \cdot 23 \cdot 37^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-32670166345301477757058771456 $ | = | $-1 \cdot 2^{9} \cdot 7^{9} \cdot 23^{3} \cdot 37^{9} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( -\frac{5282409060555942439635337}{12733300881458665984} \) | = | $-1 \cdot 2^{-9} \cdot 7^{-9} \cdot 23^{-3} \cdot 37^{-3} \cdot 61^{3} \cdot 199^{3} \cdot 14347^{3}$ | 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: | $4.3504699426419989816746663197\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $2.5450109863198867594906184842\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $0.9828352421179953\dots$ | |||
| Szpiro ratio: | $6.047533551496946\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $2.0680831199470976407897520607\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.0089904263420293812685323359649\dots$ | 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: | $ 324 $ = $ 3^{2}\cdot3^{2}\cdot1\cdot2^{2} $ | 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) $ ≈ $ 6.0241154627414972388696975755 $ | 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
$\displaystyle 6.024115463 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.008990 \cdot 2.068083 \cdot 324}{1^2} \approx 6.024115463$
Modular invariants
Modular form 440818.2.a.o
For more coefficients, see the Downloads section to the right.
| Modular degree: | 760586112 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $9$ | $I_{9}$ | Split multiplicative | -1 | 1 | 9 | 9 |
| $7$ | $9$ | $I_{9}$ | Split multiplicative | -1 | 1 | 9 | 9 |
| $23$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
| $37$ | $4$ | $I_{3}^{*}$ | Additive | 1 | 2 | 9 | 3 |
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 |
|---|---|---|
| $3$ | 3Cs | 3.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 428904 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 23 \cdot 37 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 214443 & 428896 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 82 \end{array}\right),\left(\begin{array}{rr} 279730 & 9 \\ 391599 & 428896 \end{array}\right),\left(\begin{array}{rr} 367642 & 9 \\ 183807 & 428896 \end{array}\right),\left(\begin{array}{rr} 34772 & 428895 \\ 243441 & 20 \end{array}\right),\left(\begin{array}{rr} 428887 & 18 \\ 428886 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 214462 & 9 \\ 107217 & 428896 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 321689 & 214470 \\ 214290 & 130789 \end{array}\right)$.
The torsion field $K:=\Q(E[428904])$ is a degree-$40702423547076673536$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/428904\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 440818o
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 11914e2, its twist by $37$.
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.