Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, -199, 510]); // or
magma: E := EllipticCurve("289a1");
sage: E = EllipticCurve([1, -1, 1, -199, 510]) # or
sage: E = EllipticCurve("289a1")
gp: E = ellinit([1, -1, 1, -199, 510]) \\ or
gp: E = ellinit("289a1")

$$y^2 + x y + y = x^{3} - x^{2} - 199 x + 510$$

## Mordell-Weil group structure

$$\Z\times \Z/{4}\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

 $$P$$ = $$\left(-12, 38\right)$$ $$\hat{h}(P)$$ ≈ 3.03525617773

## Torsion generators

magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp: elltors(E)

$$\left(30, 129\right)$$

## Integral points

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

$$\left(-12, 38\right)$$, $$\left(13, -7\right)$$, $$\left(30, 129\right)$$

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E) Conductor: $$289$$ = $$17^{2}$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$410338673$$ = $$17^{7}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{35937}{17}$$ = $$3^{3} \cdot 11^{3} \cdot 17^{-1}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E); sage: E.rank() Rank: $$1$$ magma: Regulator(E); sage: E.regulator() Regulator: $$3.03525617773$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega Real period: $$1.50088782003$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$4$$  = $$2^{2}$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E) Torsion order: $$4$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form289.2.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)

$$q - q^{2} - q^{4} + 2q^{5} - 4q^{7} + 3q^{8} - 3q^{9} - 2q^{10} - 2q^{13} + 4q^{14} - q^{16} + 3q^{18} - 4q^{19} + O(q^{20})$$

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

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 72 $$\Gamma_0(N)$$-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/factorial(ar)

$$L'(E,1)$$ ≈ $$1.13889475696$$

## 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$$(j)_{-}$$
$$17$$ $$4$$ $$I_1^{*}$$ Additive 1 2 7 1

## Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X242h.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^5\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 3 \\ 20 & 5 \end{array}\right)$ and has index 96.

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 $$p$$ except those listed.

prime Image of Galois representation
$$2$$ B

## $p$-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 ordinary ss ordinary ordinary ss ordinary add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ? 1,1 1 1 1,1 1 - 1 1 1 1 1 1 1 1,1 ? 0,0 0 0 0,0 0 - 0 0 0 0 0 0 0 0,0

An entry ? indicates that the invariants have not yet been computed.

An entry - indicates that the invariants are not computed because the reduction is additive.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 289.a consists of 4 curves linked by isogenies of degrees dividing 4.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{17})$$ $$\Z/2\Z \times \Z/4\Z$$ 2.2.17.1-17.1-a4
4 4.4.4913.1 $$\Z/2\Z \times \Z/8\Z$$ 4.4.4913.1-17.1-a8

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.