Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 0, -408, 1588]); // or

magma: E := EllipticCurve("19600be1");

sage: E = EllipticCurve([0, 1, 0, -408, 1588]) # or

sage: E = EllipticCurve("19600be1")

gp: E = ellinit([0, 1, 0, -408, 1588]) \\ or

gp: E = ellinit("19600be1")

$$y^2 = x^{3} + x^{2} - 408 x + 1588$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-22, 20\right)$$ $$\left(-12, 70\right)$$ $$\hat{h}(P)$$ ≈ 1.18725666317 0.248685904005

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$(-22,\pm 20)$$, $$(-12,\pm 70)$$, $$(2,\pm 28)$$, $$(3,\pm 20)$$, $$(4,\pm 6)$$, $$(18,\pm 20)$$, $$(23,\pm 70)$$, $$(44,\pm 266)$$, $$(58,\pm 420)$$, $$(114,\pm 1204)$$, $$(188,\pm 2570)$$, $$(282,\pm 4732)$$, $$(668,\pm 17270)$$, $$(2298,\pm 110180)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E) Conductor: $$19600$$ = $$2^{4} \cdot 5^{2} \cdot 7^{2}$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$3073280000$$ = $$2^{11} \cdot 5^{4} \cdot 7^{4}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$2450$$ = $$2 \cdot 5^{2} \cdot 7^{2}$$ 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: $$0.11483524367$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega Real period: $$1.28929879872$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$36$$  = $$2^{2}\cdot3\cdot3$$ 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 19600.2.a.l

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 9792 $$\Gamma_0(N)$$-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)

$$L^{(2)}(E,1)/2!$$ ≈ $$5.3300499017$$

## 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$$(j)_{-}$$
$$2$$ $$4$$ $$I_3^{*}$$ Additive 1 4 11 0
$$5$$ $$3$$ $$IV$$ Additive -1 2 4 0
$$7$$ $$3$$ $$IV$$ Additive 1 2 4 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 X5.

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

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 add ordinary add add ordinary ordinary ordinary ss ordinary ordinary ordinary ss ordinary ordinary ordinary - 8 - - 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

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

## Isogenies

This curve has no rational isogenies. Its isogeny class 19600.l 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 3.3.9800.1 $$\Z/2\Z$$ Not in database
6 6.6.768320000.1 $$\Z/2\Z \times \Z/2\Z$$ 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.