Properties

Label 16830bo2
Conductor 16830
Discriminant -20593781223840
j-invariant \( \frac{1434104310933}{1046272480} \)
CM no
Rank 1
Torsion Structure \(\mathrm{Trivial}\)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, 6343, -100871]); // or
 
magma: E := EllipticCurve("16830bo2");
 
sage: E = EllipticCurve([1, -1, 1, 6343, -100871]) # or
 
sage: E = EllipticCurve("16830bo2")
 
gp: E = ellinit([1, -1, 1, 6343, -100871]) \\ or
 
gp: E = ellinit("16830bo2")
 

\( y^2 + x y + y = x^{3} - x^{2} + 6343 x - 100871 \)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

magma: Generators(E);
 
sage: E.gens()
 

\(P\) =  \( \left(19, 152\right) \)
\(\hat{h}(P)\) ≈  1.97592914341

Integral points

magma: IntegralPoints(E);
 
sage: E.integral_points()
 

\( \left(19, 152\right) \)

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

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 16830 \)  =  \(2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 17\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(-20593781223840 \)  =  \(-1 \cdot 2^{5} \cdot 3^{9} \cdot 5 \cdot 11^{3} \cdot 17^{3} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( \frac{1434104310933}{1046272480} \)  =  \(2^{-5} \cdot 3^{6} \cdot 5^{-1} \cdot 7^{3} \cdot 11^{-3} \cdot 17^{-3} \cdot 179^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(1\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(1.97592914341\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.383109824004\)
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: \( 10 \)  = \( 5\cdot2\cdot1\cdot1\cdot1 \)
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: \(1\) (exact)

Modular invariants

Modular form 16830.2.a.cg

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)
 

\( q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} + q^{10} - q^{11} - 4q^{13} - q^{14} + q^{16} - q^{17} + 2q^{19} + O(q^{20}) \)

For more coefficients, see the Downloads section to the right.

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 43200
\( \Gamma_0(N) \)-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])
 

\( L'(E,1) \) ≈ \( 7.56997866377 \)

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\((j)_{-}\)
\(2\) \(5\) \( I_{5} \) Split multiplicative -1 1 5 5
\(3\) \(2\) \( III^{*} \) Additive 1 2 9 0
\(5\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1
\(11\) \(1\) \( I_{3} \) Non-split multiplicative 1 1 3 3
\(17\) \(1\) \( I_{3} \) Non-split multiplicative 1 1 3 3

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.1.2

$p$-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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type split add split ordinary nonsplit ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 2 - 2 3 1 1 1 1 1 1 1 1 1 1 1
$\mu$-invariant(s) 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 16830bo 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 \(\Q(\sqrt{-3}) \) \(\Z/3\Z\) Not in database
3 3.1.2700.1 \(\Z/3\Z\) Not in database
3.1.22440.1 \(\Z/2\Z\) Not in database
6 6.0.21870000.2 \(\Z/3\Z \times \Z/3\Z\) Not in database
6.0.11299742784000.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
6.0.1510660800.1 \(\Z/6\Z\) 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.