Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-x^2-431421397x+5369381586269\) | (homogenize, simplify) |
\(y^2z=x^3-x^2z-431421397xz^2+5369381586269z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-34945133184x+3914174340990576\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(\frac{252613}{4}, \frac{121324931}{8}\right)\) | \(\left(-7076, 2840383\right)\) |
$\hat{h}(P)$ | ≈ | $0.77637743141740108923319790091$ | $2.0686958934939223901449190237$ |
Integral points
\((-25276,\pm 353717)\), \((-9820,\pm 2942597)\), \((-7076,\pm 2840383)\), \((23668,\pm 2901059)\)
Invariants
Conductor: | \( 435344 \) | = | $2^{4} \cdot 7 \cdot 13^{2} \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-7314928862066456615512027136 $ | = | $-1 \cdot 2^{12} \cdot 7^{12} \cdot 13^{9} \cdot 23^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{449167881463536812032}{369990050199923699} \) | = | $-1 \cdot 2^{21} \cdot 7^{-12} \cdot 13^{-3} \cdot 19^{3} \cdot 23^{-3} \cdot 47^{3} \cdot 67^{3}$ | 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: | $4.0446361587135560808042583294\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.0690142994228424033602824872\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.039990952304704\dots$ | |||
Szpiro ratio: | $5.55662052523304\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.5020440351039714414642532521\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.038343991646730109331084367854\dots$ | 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: | $ 144 $ = $ 1\cdot( 2^{2} \cdot 3 )\cdot2^{2}\cdot3 $ | 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! $ ≈ $ 8.2935884066468354004465979259 $ | 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
$\displaystyle 8.293588407 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.038344 \cdot 1.502044 \cdot 144}{1^2} \approx 8.293588407$
Modular invariants
Modular form 435344.2.a.o
For more coefficients, see the Downloads section to the right.
Modular degree: | 282175488 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | not computed* (one of 3 curves in this isogeny class which might be optimal) | |
Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $II^{*}$ | Additive | -1 | 4 | 12 | 0 |
$7$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
$13$ | $4$ | $I_{3}^{*}$ | Additive | 1 | 2 | 9 | 3 |
$23$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 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 \( 75348 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 13 \cdot 23 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 37673 & 0 \\ 0 & 75347 \end{array}\right),\left(\begin{array}{rr} 75337 & 75330 \\ 37836 & 54683 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 82 \end{array}\right),\left(\begin{array}{rr} 64571 & 75330 \\ 54270 & 38297 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 37691 & 75330 \\ 37692 & 75329 \end{array}\right),\left(\begin{array}{rr} 75331 & 18 \\ 75330 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 40571 & 75330 \\ 63747 & 75185 \end{array}\right),\left(\begin{array}{rr} 16381 & 18 \\ 34407 & 163 \end{array}\right)$.
The torsion field $K:=\Q(E[75348])$ is a degree-$36588435897581568$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/75348\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 435344.o
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 2093.h2, its twist by $-52$.
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.