Minimal Weierstrass equation
magma: E := EllipticCurve([1, 1, 0, -8, 6]); // or
magma: E := EllipticCurve("786a1");
sage: E = EllipticCurve([1, 1, 0, -8, 6]) # or
sage: E = EllipticCurve("786a1")
gp: E = ellinit([1, 1, 0, -8, 6]) \\ or
gp: E = ellinit("786a1")
\( y^2 + x y = x^{3} + x^{2} - 8 x + 6 \)
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
magma: Generators(E);
sage: E.gens()
| \(P\) | = | \( \left(1, 0\right) \) |
| \(\hat{h}(P)\) | ≈ | 0.287032776977 |
Integral points
magma: IntegralPoints(E);
sage: E.integral_points()
\( \left(1, 0\right) \), \( \left(3, 3\right) \)
Note: only one of each pair $\pm P$ is listed.
Invariants
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magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
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| Conductor: | \( 786 \) | = | \(2 \cdot 3 \cdot 131\) | ||
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magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
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| Discriminant: | \(786 \) | = | \(2 \cdot 3 \cdot 131 \) | ||
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magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
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| j-invariant: | \( \frac{68417929}{786} \) | = | \(2^{-1} \cdot 3^{-1} \cdot 131^{-1} \cdot 409^{3}\) | ||
| Endomorphism ring: | \(\Z\) | (no Complex Multiplication) | |||
| Sato-Tate Group: | $\mathrm{SU}(2)$ | ||||
BSD invariants
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magma: Rank(E);
sage: E.rank()
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| Rank: | \(1\) | ||
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magma: Regulator(E);
sage: E.regulator()
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| Regulator: | \(0.287032776977\) | ||
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magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
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| Real period: | \(5.05815649234\) | ||
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magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
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| Tamagawa product: | \( 1 \) = \( 1\cdot1\cdot1 \) | ||
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magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
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| Torsion order: | \(1\) | ||
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magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
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| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 786.2.a.c
magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
\( q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{7} - q^{8} + q^{9} + q^{10} - q^{11} - q^{12} + 4q^{13} + q^{14} + q^{15} + q^{16} + 3q^{17} - q^{18} + O(q^{20}) \)
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magma: ModularDegree(E);
sage: E.modular_degree()
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| Modular degree: | 40 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
sage: r = E.rank();
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp: ar = ellanalyticrank(E);
gp: ar[2]/factorial(ar[1])
\( L'(E,1) \) ≈ \( 1.45185670438 \)
Local data
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(E)[5]
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(1\) | \( I_{1} \) | Non-split multiplicative | 1 | 1 | 1 | 1 |
| \(3\) | \(1\) | \( I_{1} \) | Non-split multiplicative | 1 | 1 | 1 | 1 |
| \(131\) | \(1\) | \( I_{1} \) | Non-split multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The 2-adic representation attached to this elliptic curve is surjective.
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
sage: rho = E.galois_representation();
sage: [rho.image_type(p) for p in rho.non_surjective()]
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) .
$p$-adic data
$p$-adic regulators
sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]
Note: \(p\)-adic regulator data only exists for primes \(p\ge5\) of good ordinary reduction.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 131 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | nonsplit | ordinary | ordinary | ordinary | ordinary | ordinary | ss | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | nonsplit |
| $\lambda$-invariant(s) | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 3 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Isogenies
This curve has no rational isogenies. Its isogeny class 786a consists of this curve only.
Growth of torsion in number fields
The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base-change curve |
|---|---|---|---|
| 3 | 3.3.3144.1 | \(\Z/2\Z\) | Not in database |
| 6 | 6.6.31077609984.1 | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.