Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 0, -2, 0]); // or
magma: E := EllipticCurve("102a1");
sage: E = EllipticCurve([1, 1, 0, -2, 0]) # or
sage: E = EllipticCurve("102a1")
gp: E = ellinit([1, 1, 0, -2, 0]) \\ or
gp: E = ellinit("102a1")

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

## Mordell-Weil group structure

$$\Z\times \Z/{2}\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

 $$P$$ = $$\left(-1, 2\right)$$ $$\hat{h}(P)$$ ≈ 0.143253892941

## Torsion generators

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

$$\left(0, 0\right)$$

## Integral points

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

$$\left(-2, 2\right)$$, $$\left(-1, 2\right)$$, $$\left(0, 0\right)$$, $$\left(1, 0\right)$$, $$\left(2, 2\right)$$, $$\left(8, 20\right)$$, $$\left(9, 24\right)$$, $$\left(2738, 141932\right)$$

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E) Conductor: $$102$$ = $$2 \cdot 3 \cdot 17$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$612$$ = $$2^{2} \cdot 3^{2} \cdot 17$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{1771561}{612}$$ = $$2^{-2} \cdot 3^{-2} \cdot 11^{6} \cdot 17^{-1}$$ 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: $$0.143253892941$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega Real period: $$4.72786382354$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$4$$  = $$2\cdot2\cdot1$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E) Torsion order: $$2$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form102.2.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)

$$q - q^{2} - q^{3} + q^{4} - 4q^{5} + q^{6} - 2q^{7} - q^{8} + q^{9} + 4q^{10} - q^{12} - 6q^{13} + 2q^{14} + 4q^{15} + q^{16} - q^{17} - q^{18} + 4q^{19} + O(q^{20})$$

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 8 $$\Gamma_0(N)$$-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)

$$L'(E,1)$$ ≈ $$0.677284898017$$

## 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$17$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 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 X15.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 2 & 1 \end{array}\right)$ and has index 6.

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 nonsplit nonsplit ordinary ordinary ss ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary 1 1 1 1 1,1 1 1 1 3 1 1 1 1 1 1 0 0 0 0 0,0 0 0 0 0 0 0 0 0 0 0

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 102a consists of 2 curves linked by isogenies of degree 2.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{17})$$ $$\Z/2\Z \times \Z/2\Z$$ 2.2.17.1-612.1-a2
4 4.0.1088.1 $$\Z/4\Z$$ 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.

This is the elliptic curve $E$ associated to the [Somos-5 sequence] $\{a(n)\}$. Let $T$ be the $2$-torsion point $(0,0)$, and $P$ the point $(2,2)$ such that $E(\Q) = \Z P \oplus \{0, T\}$. Then the $x$- and $y$-coordinates of $nP+T$ have denominators $d_n^2$ and $d_n^3$ where $$d_n = 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217$$ for $1 \leq n \leq 10$, and $d_n = a(n+2)$ in general, satisfying the Somos-5 recurrence $$d_n d_{n+5} = d_{n+1} d_{n+4} + d_{n+2} d_{n+3}.$$ Thus the regulator of $E$, which is the canonical height $\hat h(P) = 0.143\ldots$, controls the growth of the $a(n)$: asymptotically $\log a_n \sim \frac12 \hat h(P) n^2$.