Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-2823440x-3087704968\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-2823440xz^2-3087704968z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-45175035x-197658292970\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(\frac{243561}{25}, \frac{117583076}{125}\right)\) | \(\left(\frac{51954}{25}, \frac{378548}{125}\right)\) |
$\hat{h}(P)$ | ≈ | $4.1386277613451628846860407575$ | $5.9354361271225443688935715017$ |
Integral points
None
Invariants
Conductor: | \( 455175 \) | = | $3^{2} \cdot 5^{2} \cdot 7 \cdot 17^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-2680030097613128576925 $ | = | $-1 \cdot 3^{17} \cdot 5^{2} \cdot 7 \cdot 17^{9} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{1151731985}{1240029} \) | = | $-1 \cdot 3^{-11} \cdot 5 \cdot 7^{-1} \cdot 613^{3}$ | 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: | $2.8065890580172238101694486766\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.13586674643134315814878479414\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9078569396125251\dots$ | |||
Szpiro ratio: | $4.392314249730676\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $22.956802446678780388113750787\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.055858085968384194116727355037\dots$ | 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: | $ 8 $ = $ 2^{2}\cdot1\cdot1\cdot2 $ | 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! $ ≈ $ 10.258584357006367355516291471 $ | 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
$\displaystyle 10.258584357 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.055858 \cdot 22.956802 \cdot 8}{1^2} \approx 10.258584357$
Modular invariants
Modular form 455175.2.a.bc
For more coefficients, see the Downloads section to the right.
Modular degree: | 17233920 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $4$ | $I_{11}^{*}$ | Additive | -1 | 2 | 17 | 11 |
$5$ | $1$ | $II$ | Additive | 1 | 2 | 2 | 0 |
$7$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$17$ | $2$ | $III^{*}$ | Additive | 1 | 2 | 9 | 0 |
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 \( 1428 = 2^{2} \cdot 3 \cdot 7 \cdot 17 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 409 & 2 \\ 409 & 3 \end{array}\right),\left(\begin{array}{rr} 953 & 2 \\ 953 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 715 & 2 \\ 715 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 1427 & 0 \end{array}\right),\left(\begin{array}{rr} 1261 & 2 \\ 1261 & 3 \end{array}\right),\left(\begin{array}{rr} 1427 & 2 \\ 1426 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[1428])$ is a degree-$363860066304$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1428\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 455175bc consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 151725cu1, its twist by $-51$.
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.