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## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, 39, -19]) # or

sage: E = EllipticCurve("162a2")

gp: E = ellinit([1, -1, 0, 39, -19]) \\ or

gp: E = ellinit("162a2")

magma: E := EllipticCurve([1, -1, 0, 39, -19]); // or

magma: E := EllipticCurve("162a2");

$$y^2 + x y = x^{3} - x^{2} + 39 x - 19$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(10, -41\right)$$ $$\hat{h}(P)$$ ≈ 0.1019782946229211

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(1, 4\right)$$, $$\left(1, -5\right)$$, $$\left(2, 7\right)$$, $$\left(2, -9\right)$$, $$\left(10, 31\right)$$, $$\left(10, -41\right)$$, $$\left(19, 76\right)$$, $$\left(19, -95\right)$$, $$\left(154, 1831\right)$$, $$\left(154, -1985\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$162$$ = $$2 \cdot 3^{4}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-3779136$$ = $$-1 \cdot 2^{6} \cdot 3^{10}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{109503}{64}$$ = $$2^{-6} \cdot 3^{2} \cdot 23^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.101978294623$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$1.46514758762$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$6$$  = $$2\cdot3$$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $$1$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form162.2.a.a

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - q^{2} + q^{4} - 3q^{5} - 4q^{7} - q^{8} + 3q^{10} - q^{13} + 4q^{14} + q^{16} - 3q^{17} - 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 36 $$\Gamma_0(N)$$-optimal: no 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/factorial(ar)

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

$$L'(E,1)$$ ≈ $$0.896479514136$$

## Local data

This elliptic curve is not semistable.

sage: E.local_data()

gp: ellglobalred(E)

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

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$2$$ $$I_{6}$$ Non-split multiplicative 1 1 6 6
$$3$$ $$3$$ $$IV^{*}$$ Additive 1 4 10 0

## 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 X20.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 3 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 3 & 0 \end{array}\right)$ and has index 8.

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 $$p$$ except those listed.

prime Image of Galois representation
$$3$$ B.1.2

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

$$p$$-adic regulators are not yet computed for curves that are not $$\Gamma_0$$-optimal.

## 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 nonsplit add ordinary ordinary ss ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary 4 - 1 1 1,1 3 1 1 1,1 1 1 1 1 1 3 0 - 0 0 0,0 0 0 0 0,0 0 0 0 0 0 0

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 $$d=$$ 3.
Its isogeny class 162.a consists of 2 curves linked by isogenies of degree 3.

## 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
2 $$\Q(\sqrt{-3})$$ $$\Z/3\Z$$ 2.0.3.1-2916.1-a2
3 3.1.243.1 $$\Z/3\Z$$ Not in database
3.1.324.1 $$\Z/2\Z$$ Not in database
6 6.0.177147.2 $$\Z/3\Z \times \Z/3\Z$$ Not in database
6.0.314928.2 $$\Z/12\Z$$ Not in database
6.0.419904.2 $$\Z/2\Z \times \Z/2\Z$$ Not in database
6.2.1259712.2 $$\Z/4\Z$$ 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.