# Properties

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

# Related objects

Show commands for: Magma / Pari/GP / SageMath

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

## Minimal Weierstrass equation

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")

magma: E := EllipticCurve([0, 1, 1, -2, 0]); // or

magma: E := EllipticCurve("389a1");

$$y^2 + y = x^{3} + x^{2} - 2 x$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-1, 1\right)$$ $$\left(0, 0\right)$$ $$\hat{h}(P)$$ ≈ 0.6866670833055866 0.32700077365160496

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-2, 0\right)$$, $$\left(-2, -1\right)$$, $$\left(-1, 1\right)$$, $$\left(-1, -2\right)$$, $$\left(0, 0\right)$$, $$\left(0, -1\right)$$, $$\left(1, 0\right)$$, $$\left(1, -1\right)$$, $$\left(3, 5\right)$$, $$\left(3, -6\right)$$, $$\left(4, 8\right)$$, $$\left(4, -9\right)$$, $$\left(6, 15\right)$$, $$\left(6, -16\right)$$, $$\left(39, 246\right)$$, $$\left(39, -247\right)$$, $$\left(133, 1539\right)$$, $$\left(133, -1540\right)$$, $$\left(188, 2584\right)$$, $$\left(188, -2585\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$389$$ = $$389$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$389$$ = $$389$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); 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

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

## Modular invariants

#### Modular form389.2.a.a

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

magma: ModularForm(E);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 40 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar[2]/factorial(ar[1])

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

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

## Local data

This elliptic curve is semistable.

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

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.

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

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

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 389 ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary split 2,3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 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.