Properties

 Label 19600bx2 Conductor 19600 Discriminant -38416000000000 j-invariant $$-\frac{5452947409}{250}$$ CM no Rank 2 Torsion Structure $$\mathrm{Trivial}$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 0, -196408, 33439188]); // or

magma: E := EllipticCurve("19600bx2");

sage: E = EllipticCurve([0, 1, 0, -196408, 33439188]) # or

sage: E = EllipticCurve("19600bx2")

gp: E = ellinit([0, 1, 0, -196408, 33439188]) \\ or

gp: E = ellinit("19600bx2")

$$y^2 = x^{3} + x^{2} - 196408 x + 33439188$$

Mordell-Weil group structure

$$\Z^2$$

Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-397, 7000\right)$$ $$\left(-82, 7000\right)$$ $$\hat{h}(P)$$ ≈ 2.40657943324 1.57791733978

Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$(-397,\pm 7000)$$, $$(-82,\pm 7000)$$, $$(-68,\pm 6818)$$, $$(174,\pm 2136)$$, $$(228,\pm 750)$$, $$(247,\pm 238)$$, $$(254,\pm 56)$$, $$(268,\pm 350)$$, $$(318,\pm 1800)$$, $$(478,\pm 7000)$$, $$(1262,\pm 42392)$$, $$(2228,\pm 103250)$$, $$(974318,\pm 961725800)$$

Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$19600$$ = $$2^{4} \cdot 5^{2} \cdot 7^{2}$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-38416000000000$$ = $$-1 \cdot 2^{13} \cdot 5^{9} \cdot 7^{4}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{5452947409}{250}$$ = $$-1 \cdot 2^{-1} \cdot 5^{-3} \cdot 7^{2} \cdot 13^{3} \cdot 37^{3}$$ 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: $$0.180588002078$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.609907114066$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$48$$  = $$2^{2}\cdot2^{2}\cdot3$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$1$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (rounded)

Modular invariants

Modular form 19600.2.a.o

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

$$q - 2q^{3} + q^{9} - 3q^{11} - 5q^{13} - 6q^{17} + q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 103680 $$\Gamma_0(N)$$-optimal: no 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[2]/factorial(ar[1])

$$L^{(2)}(E,1)/2!$$ ≈ $$5.28681154474$$

Local data

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

sage: E.local_data()

gp: ellglobalred(E)[5]

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$4$$ $$I_5^{*}$$ Additive -1 4 13 1
$$5$$ $$4$$ $$I_3^{*}$$ Additive 1 2 9 3
$$7$$ $$3$$ $$IV$$ Additive 1 2 4 0

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 $$p$$ except those listed.

prime Image of Galois representation
$$3$$ B

$p$-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 add ordinary add add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary - 4 - - 2 2 2 4 2 2 2 2 2 2 2 - 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 19600bx 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 $$\Q(\sqrt{15})$$ $$\Z/3\Z$$ Not in database
3 3.1.1960.1 $$\Z/2\Z$$ Not in database
6 6.0.153664000.2 $$\Z/2\Z \times \Z/2\Z$$ Not in database
6.2.8297856000.14 $$\Z/6\Z$$ Not in database
6.0.56010528000.3 $$\Z/3\Z$$ 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.