Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+x^2-839500x-293066000\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+x^2z-839500xz^2-293066000z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1087992675x-13656967409250\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(-555, 1790\right)\) | \(\left(17595, 2321965\right)\) |
$\hat{h}(P)$ | ≈ | $0.74343788539593712538785150737$ | $3.0282200552344371315357910950$ |
Integral points
\( \left(-555, 1790\right) \), \( \left(-555, -1235\right) \), \( \left(-529, 2093\right) \), \( \left(-529, -1564\right) \), \( \left(1865, 67130\right) \), \( \left(1865, -68995\right) \), \( \left(3795, 224365\right) \), \( \left(3795, -228160\right) \), \( \left(17595, 2321965\right) \), \( \left(17595, -2339560\right) \)
Invariants
Conductor: | \( 417450 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 11^{2} \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $850468860367500000 $ | = | $2^{5} \cdot 3 \cdot 5^{7} \cdot 11^{8} \cdot 23^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{19535526241}{253920} \) | = | $2^{-5} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{3} \cdot 11 \cdot 23^{-2} \cdot 173^{3}$ | 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: | $2.2484331734677080333899046069\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.15488263128158918328510411169\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.8502097914709049\dots$ | |||
Szpiro ratio: | $4.059311905394376\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.1598259585101008386661603125\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.15784681663684753321892670564\dots$ | 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: | $ 24 $ = $ 1\cdot1\cdot2^{2}\cdot3\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! $ ≈ $ 8.1821196489707365219618739719 $ | 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
$\displaystyle 8.182119649 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.157847 \cdot 2.159826 \cdot 24}{1^2} \approx 8.182119649$
Modular invariants
Modular form 417450.2.a.a
For more coefficients, see the Downloads section to the right.
Modular degree: | 11151360 | 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 |
$3$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$5$ | $4$ | $I_{1}^{*}$ | Additive | 1 | 2 | 7 | 1 |
$11$ | $3$ | $IV^{*}$ | Additive | -1 | 2 | 8 | 0 |
$23$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
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 \( 120 = 2^{3} \cdot 3 \cdot 5 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 119 & 2 \\ 118 & 3 \end{array}\right),\left(\begin{array}{rr} 61 & 2 \\ 61 & 3 \end{array}\right),\left(\begin{array}{rr} 31 & 2 \\ 31 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 119 & 0 \end{array}\right),\left(\begin{array}{rr} 97 & 2 \\ 97 & 3 \end{array}\right),\left(\begin{array}{rr} 41 & 2 \\ 41 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[120])$ is a degree-$17694720$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 417450a consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 83490bh1, its twist by $-55$.
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.