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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()

 $$P$$ = $$\left(-1, 1\right)$$ $$\left(0, 0\right)$$ $$\hat{h}(P)$$ ≈ 0.686667083306 0.327000773652

## 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) 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 Real period: $$4.98042512171$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$1$$  = $$1$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E) Torsion order: $$1$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

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

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy)/(2*xy+E.a1*xy+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.

#### Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
40 . This curve is $$\Gamma_0(N)$$-optimal.

#### 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/factorial(ar)

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

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(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.

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

 $p$ 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.