Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-8740x-701866\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-8740xz^2-701866z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-11326419x-32712269202\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(136, 725\right)\) | \(\left(1723, 70553\right)\) |
$\hat{h}(P)$ | ≈ | $1.2315208798093052093307095611$ | $2.1226967104493664403836644538$ |
Integral points
\( \left(121, 65\right) \), \( \left(121, -187\right) \), \( \left(136, 725\right) \), \( \left(136, -862\right) \), \( \left(211, 2513\right) \), \( \left(211, -2725\right) \), \( \left(598, 14123\right) \), \( \left(598, -14722\right) \), \( \left(1723, 70553\right) \), \( \left(1723, -72277\right) \)
Invariants
Conductor: | \( 415794 \) | = | $2 \cdot 3 \cdot 23^{2} \cdot 131$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-169647352363332 $ | = | $-1 \cdot 2^{2} \cdot 3^{7} \cdot 23^{6} \cdot 131 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{498677257}{1145988} \) | = | $-1 \cdot 2^{-2} \cdot 3^{-7} \cdot 13^{3} \cdot 61^{3} \cdot 131^{-1}$ | 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: | $1.4188283131463753527645595572\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.14891879481819949263881685871\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.8953888482780743\dots$ | |||
Szpiro ratio: | $3.1260036508589137\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.6137935939435119642280341563\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.23074047050374391943041554246\dots$ | 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: | $ 28 $ = $ 2\cdot7\cdot2\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^{(2)}(E,1)/2! $ ≈ $ 16.887022982653536542810863610 $ | 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
$\displaystyle 16.887022983 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.230740 \cdot 2.613794 \cdot 28}{1^2} \approx 16.887022983$
Modular invariants
Modular form 415794.2.a.v
For more coefficients, see the Downloads section to the right.
Modular degree: | 2069760 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$3$ | $7$ | $I_{7}$ | Split multiplicative | -1 | 1 | 7 | 7 |
$23$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$131$ | $1$ | $I_{1}$ | 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 \( 1572 = 2^{2} \cdot 3 \cdot 131 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 133 & 2 \\ 133 & 3 \end{array}\right),\left(\begin{array}{rr} 1049 & 2 \\ 1049 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1571 & 2 \\ 1570 & 3 \end{array}\right),\left(\begin{array}{rr} 787 & 2 \\ 787 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 1571 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[1572])$ is a degree-$673308979200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1572\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 415794.v consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 786.g1, its twist by $-23$.
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.