Properties

 Label 10830.bb2 Conductor $10830$ Discriminant $-95004009000$ j-invariant $$\frac{129205871}{729000}$$ CM no Rank $1$ Torsion structure $$\Z/{3}\Z$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, 534, -14004]) # or

sage: E = EllipticCurve("10830.bb2")

gp: E = ellinit([1, 0, 0, 534, -14004]) \\ or

gp: E = ellinit("10830.bb2")

magma: E := EllipticCurve([1, 0, 0, 534, -14004]); // or

magma: E := EllipticCurve("10830.bb2");

$$y^2+xy=x^3+534x-14004$$

Mordell-Weil group structure

$$\Z\times \Z/{3}\Z$$

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(18, 30\right)$$ $$\hat{h}(P)$$ ≈ $2.2694725574431332951688486008$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(30, 156\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(18, 30\right)$$, $$\left(18, -48\right)$$, $$\left(30, 156\right)$$, $$\left(30, -186\right)$$, $$\left(258, 4032\right)$$, $$\left(258, -4290\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$10830$$ = $$2 \cdot 3 \cdot 5 \cdot 19^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-95004009000$$ = $$-1 \cdot 2^{3} \cdot 3^{6} \cdot 5^{3} \cdot 19^{4}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{129205871}{729000}$$ = $$2^{-3} \cdot 3^{-6} \cdot 5^{-3} \cdot 19^{2} \cdot 71^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$2.2694725574431332951688486008$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.53557412382805522216665772121$$ 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: $$54$$  = $$3\cdot( 2 \cdot 3 )\cdot1\cdot3$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$3$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

Modular invariants

Modular form 10830.2.a.bb

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);

$$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} - q^{7} + q^{8} + q^{9} - q^{10} - 3q^{11} + q^{12} + 2q^{13} - q^{14} - q^{15} + q^{16} - 6q^{17} + q^{18} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 12960 $$\Gamma_0(N)$$-optimal: yes 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);

$$L'(E,1)$$ ≈ $$7.2928246590265310381211770649072705696$$

Local data

This elliptic curve is semistable. There are 4 primes of bad reduction:

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$$(j)_{-}$$
$$2$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$3$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$5$$ $$1$$ $$I_{3}$$ Non-split multiplicative 1 1 3 3
$$19$$ $$3$$ $$IV$$ Additive 1 2 4 0

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.1.1

$p$-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 $$p\ge 5$$ 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 split split nonsplit ordinary ordinary ordinary ordinary add ordinary ss ordinary ordinary ordinary ordinary ss 2 4 1 1 1 1 1 - 1 1,1 1 1 1 1 1,1 0 0 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 non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 10830.bb consists of 2 curves linked by isogenies of degree 3.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.14440.1 $$\Z/6\Z$$ Not in database $6$ 6.0.8340544000.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database $6$ 6.0.3518667.2 $$\Z/3\Z \times \Z/3\Z$$ Not in database $9$ 9.3.43203646156932288.4 $$\Z/9\Z$$ Not in database $12$ Deg 12 $$\Z/12\Z$$ Not in database $18$ 18.0.178440919252906995560448000000.2 $$\Z/3\Z \times \Z/6\Z$$ Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.