Properties

 Label 193600gd2 Conductor 193600 Discriminant -123624367901900800 j-invariant $$-\frac{53969305}{10648}$$ CM no Rank 2 Torsion Structure $$\mathrm{Trivial}$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -178273, 33489663]) # or

sage: E = EllipticCurve("193600gd2")

gp: E = ellinit([0, 1, 0, -178273, 33489663]) \\ or

gp: E = ellinit("193600gd2")

magma: E := EllipticCurve([0, 1, 0, -178273, 33489663]); // or

magma: E := EllipticCurve("193600gd2");

$$y^2 = x^{3} + x^{2} - 178273 x + 33489663$$

Mordell-Weil group structure

$$\Z^2$$

Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(29, 5324\right)$$ $$\left(\frac{2923}{9}, -\frac{85184}{27}\right)$$ $$\hat{h}(P)$$ ≈ 1.2067376207226046 1.3600230183134934

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-301,\pm 7744)$$, $$(29,\pm 5324)$$, $$(467,\pm 7232)$$, $$(637,\pm 13372)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$193600$$ = $$2^{6} \cdot 5^{2} \cdot 11^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-123624367901900800$$ = $$-1 \cdot 2^{21} \cdot 5^{2} \cdot 11^{9}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{53969305}{10648}$$ = $$-1 \cdot 2^{-3} \cdot 5 \cdot 11^{-3} \cdot 13^{3} \cdot 17^{3}$$ Endomorphism Ring: $$\Z$$ Geometric Endomorphism Ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1.33798708751$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.316933502586$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$16$$  = $$2^{2}\cdot1\cdot2^{2}$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$1$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (rounded)

Modular invariants

Modular form 193600.2.a.s

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

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

magma: ModularForm(E);

$$q - 2q^{3} - 4q^{7} + q^{9} - 5q^{13} - 7q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2488320 $$\Gamma_0(N)$$-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[2]/factorial(ar[1])

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

$$L^{(2)}(E,1)/2!$$ ≈ $$6.78484694498$$

Local data

This elliptic curve is not semistable.

sage: E.local_data()

gp: ellglobalred(E)[5]

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

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$4$$ $$I_11^{*}$$ Additive 1 6 21 3
$$5$$ $$1$$ $$II$$ Additive 1 2 2 0
$$11$$ $$4$$ $$I_3^{*}$$ Additive -1 2 9 3

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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

$p$-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.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 193600gd 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 $$\Q(\sqrt{330})$$ $$\Z/3\Z$$ Not in database
3 3.1.2200.1 $$\Z/2\Z$$ Not in database
6 6.0.425920000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database
6.2.57499200000.1 $$\Z/6\Z$$ Not in database
6.0.1552478400000.9 $$\Z/3\Z$$ 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.