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Properties

Label 389.a1
Conductor 389
Discriminant 389
j-invariant \( \frac{1404928}{389} \)
CM no
Rank 2
Torsion Structure \(\mathrm{Trivial}\)

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This elliptic curve has smallest conductor among elliptic curves over $\Q$ of rank 2.

Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 1, -2, 0]); // or
 
magma: E := EllipticCurve("389a1");
 
sage: E = EllipticCurve([0, 1, 1, -2, 0]) # or
 
sage: E = EllipticCurve("389a1")
 
gp: E = ellinit([0, 1, 1, -2, 0]) \\ or
 
gp: E = ellinit("389a1")
 

\( y^2 + y = x^{3} + x^{2} - 2 x \)

Mordell-Weil group structure

\(\Z^2\)

Infinite order Mordell-Weil generators and heights

magma: Generators(E);
 
sage: E.gens()
 

\(P\) =  \( \left(-1, 1\right) \)\( \left(0, 0\right) \)
\(\hat{h}(P)\) ≈  0.6866670833060.327000773652

Integral points

magma: IntegralPoints(E);
 
sage: E.integral_points()
 

\( \left(-2, 0\right) \), \( \left(-1, 1\right) \), \( \left(0, 0\right) \), \( \left(1, 0\right) \), \( \left(3, 5\right) \), \( \left(4, 8\right) \), \( \left(6, 15\right) \), \( \left(39, 246\right) \), \( \left(133, 1539\right) \), \( \left(188, 2584\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 389 \)  =  \(389\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(389 \)  =  \(389 \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( \frac{1404928}{389} \)  =  \(2^{12} \cdot 7^{3} \cdot 389^{-1}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(2\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(0.152460177943\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(4.98042512171\)
magma: TamagawaNumbers(E);
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Tamagawa product: \( 1 \)  = \( 1 \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(1\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 389.2.a.a

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)
 

\( q - 2q^{2} - 2q^{3} + 2q^{4} - 3q^{5} + 4q^{6} - 5q^{7} + q^{9} + 6q^{10} - 4q^{11} - 4q^{12} - 3q^{13} + 10q^{14} + 6q^{15} - 4q^{16} - 6q^{17} - 2q^{18} + 5q^{19} + O(q^{20}) \)

For more coefficients, see the Downloads section to the right.

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
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])
 

\( L^{(2)}(E,1)/2! \) ≈ \( 0.759316500288 \)

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\((j)_{-}\)
\(389\) \(1\) \( I_{1} \) 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 47 389
Reduction type ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary split
$\lambda$-invariant(s) 2,3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3
$\mu$-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 389.a 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 3.3.1556.1 \(\Z/2\Z\) Not in database
6 6.6.941821904.1 \(\Z/2\Z \times \Z/2\Z\) 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.