Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, -1, 0, -2169, 39613]); // or

magma: E := EllipticCurve("40616b1");

sage: E = EllipticCurve([0, -1, 0, -2169, 39613]) # or

sage: E = EllipticCurve("40616b1")

gp: E = ellinit([0, -1, 0, -2169, 39613]) \\ or

gp: E = ellinit("40616b1")

$$y^2 = x^{3} - x^{2} - 2169 x + 39613$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(27, 2\right)$$ $$\left(3, 182\right)$$ $$\hat{h}(P)$$ ≈ 0.378965134929 2.77844157493

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-29, 278\right)$$, $$\left(3, 182\right)$$, $$\left(19, 70\right)$$, $$\left(27, 2\right)$$, $$\left(28, 7\right)$$, $$\left(36, 83\right)$$

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

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E) Conductor: $$40616$$ = $$2^{3} \cdot 5077$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$1299712$$ = $$2^{8} \cdot 5077$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{4410243976192}{5077}$$ = $$2^{10} \cdot 1627^{3} \cdot 5077^{-1}$$ 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: $$0.737956175488$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega Real period: $$2.29158372411$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$4$$  = $$2^{2}\cdot1$$ 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 form 40616.2.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)

$$q - q^{3} - 2q^{5} - 2q^{9} - 2q^{11} + 2q^{13} + 2q^{15} + 4q^{17} - 5q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 11904 $$\Gamma_0(N)$$-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)

$$L^{(2)}(E,1)/2!$$ ≈ $$6.76435344342$$

## 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$$(j)_{-}$$
$$2$$ $$4$$ $$I_1^{*}$$ Additive 1 3 8 0
$$5077$$ $$1$$ $$I_{1}$$ 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 5077 add ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary split - 2 2 2,2 2 2 2 2 2 2 2 2 2 2 2 ? - 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 no rational isogenies. Its isogeny class 40616b 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 3.3.20308.1 $$\Z/2\Z$$ Not in database
6 6.6.2093830264528.1 $$\Z/2\Z \times \Z/2\Z$$ 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.