Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-890280x+322561696\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-890280xz^2+322561696z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-14244483x+20629704062\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-943, 18440\right)\) |
$\hat{h}(P)$ | ≈ | $1.2163277633792095850028567376$ |
Integral points
\( \left(-943, 18440\right) \), \( \left(-943, -17497\right) \), \( \left(4337, 277160\right) \), \( \left(4337, -281497\right) \)
Invariants
Conductor: | \( 402930 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 11^{2} \cdot 37$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $274755987659561760 $ | = | $2^{5} \cdot 3^{9} \cdot 5 \cdot 11^{9} \cdot 37 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{60425492474521}{212747040} \) | = | $2^{-5} \cdot 3^{-3} \cdot 5^{-1} \cdot 11^{-3} \cdot 37^{-1} \cdot 39241^{3}$ | 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: | $2.2084455427023353908962887653\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.46019176196909527316769435785\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.8851284349916992\dots$ | |||
Szpiro ratio: | $4.084097433278641\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.2163277633792095850028567376\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.31061236392347212608540721464\dots$ | 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: | $ 16 $ = $ 1\cdot2^{2}\cdot1\cdot2^{2}\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: | $1$ | 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) $ ≈ $ 6.0449030702234550389518956645 $ | 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
$\displaystyle 6.044903070 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.310612 \cdot 1.216328 \cdot 16}{1^2} \approx 6.044903070$
Modular invariants
Modular form 402930.2.a.o
For more coefficients, see the Downloads section to the right.
Modular degree: | 5529600 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{5}$ | Non-split multiplicative | 1 | 1 | 5 | 5 |
$3$ | $4$ | $I_{3}^{*}$ | Additive | -1 | 2 | 9 | 3 |
$5$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$11$ | $4$ | $I_{3}^{*}$ | Additive | -1 | 2 | 9 | 3 |
$37$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 48840 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 37 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 35521 & 2 \\ 35521 & 3 \end{array}\right),\left(\begin{array}{rr} 32561 & 2 \\ 32561 & 3 \end{array}\right),\left(\begin{array}{rr} 48839 & 2 \\ 48838 & 3 \end{array}\right),\left(\begin{array}{rr} 19537 & 2 \\ 19537 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 48839 & 0 \end{array}\right),\left(\begin{array}{rr} 3961 & 2 \\ 3961 & 3 \end{array}\right),\left(\begin{array}{rr} 24421 & 2 \\ 24421 & 3 \end{array}\right),\left(\begin{array}{rr} 36631 & 2 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[48840])$ is a degree-$425606202261504000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/48840\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 402930o consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 12210i1, its twist by $33$.
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.