Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+y=x^3-x^2+892940x+2079273822\) | (homogenize, simplify) |
\(y^2z+yz^2=x^3-x^2z+892940xz^2+2079273822z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+1157249808x+97024486450320\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(-\frac{2811}{4}, \frac{265833}{8}\right)\) | \(\left(5339, 398755\right)\) |
$\hat{h}(P)$ | ≈ | $1.5082586081139974293576231847$ | $1.8561823238675724473804671958$ |
Integral points
\( \left(-975, 16758\right) \), \( \left(-975, -16759\right) \), \( \left(477, 51122\right) \), \( \left(477, -51123\right) \), \( \left(5339, 398755\right) \), \( \left(5339, -398756\right) \)
Invariants
Conductor: | \( 429429 \) | = | $3 \cdot 7 \cdot 11^{2} \cdot 13^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-1913801325641316474363 $ | = | $-1 \cdot 3^{3} \cdot 7^{3} \cdot 11^{7} \cdot 13^{9} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{9208180736}{223810587} \) | = | $2^{12} \cdot 3^{-3} \cdot 7^{-3} \cdot 11^{-1} \cdot 13^{-3} \cdot 131^{3}$ | 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: | $2.7638116743101331901893896742\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.28238935918017955013167416444\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.4045289642356318590717894303\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.11096572634367079953395725013\dots$ | 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\cdot1\cdot2^{2}\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^{(2)}(E,1)/2! $ ≈ $ 4.2691248484928210552327759860 $ | 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
$\displaystyle 4.269124848 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.110966 \cdot 2.404529 \cdot 16}{1^2} \approx 4.269124848$
Modular invariants
Modular form 429429.2.a.a
For more coefficients, see the Downloads section to the right.
Modular degree: | 47900160 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
$7$ | $1$ | $I_{3}$ | Non-split multiplicative | 1 | 1 | 3 | 3 |
$11$ | $4$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
$13$ | $4$ | $I_{3}^{*}$ | Additive | 1 | 2 | 9 | 3 |
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 \( 6006 = 2 \cdot 3 \cdot 7 \cdot 11 \cdot 13 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 4369 & 2 \\ 4369 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2003 & 2 \\ 2003 & 3 \end{array}\right),\left(\begin{array}{rr} 3433 & 2 \\ 3433 & 3 \end{array}\right),\left(\begin{array}{rr} 6005 & 2 \\ 6004 & 3 \end{array}\right),\left(\begin{array}{rr} 925 & 2 \\ 925 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 6005 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[6006])$ is a degree-$100429391462400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6006\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 429429a consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 3003b1, its twist by $-143$.
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.