Properties

Label 86190bm8
Conductor 86190
Discriminant 498205345574437312500
j-invariant \( \frac{31664865542564944883878115208137569}{103216295812500} \)
CM no
Rank 0
Torsion Structure \(\Z/{2}\Z\)

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Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 1, -1113879001018, -452486084448911992]); // or
 
magma: E := EllipticCurve("86190bm8");
 
sage: E = EllipticCurve([1, 0, 1, -1113879001018, -452486084448911992]) # or
 
sage: E = EllipticCurve("86190bm8")
 
gp: E = ellinit([1, 0, 1, -1113879001018, -452486084448911992]) \\ or
 
gp: E = ellinit("86190bm8")
 

\( y^2 + x y + y = x^{3} - 1113879001018 x - 452486084448911992 \)

Mordell-Weil group structure

\(\Z/{2}\Z\)

Torsion generators

magma: TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 

\( \left(-\frac{2437353}{4}, \frac{2437349}{8}\right) \)

Integral points

magma: IntegralPoints(E);
 
sage: E.integral_points()
 
None

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 86190 \)  =  \(2 \cdot 3 \cdot 5 \cdot 13^{2} \cdot 17\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(498205345574437312500 \)  =  \(2^{2} \cdot 3^{2} \cdot 5^{6} \cdot 13^{9} \cdot 17^{4} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( \frac{31664865542564944883878115208137569}{103216295812500} \)  =  \(2^{-2} \cdot 3^{-2} \cdot 5^{-6} \cdot 11^{3} \cdot 13^{-3} \cdot 17^{-4} \cdot 433^{3} \cdot 66422003^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(0\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(1\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.00464718595068\)
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: \( 384 \)  = \( 2\cdot2\cdot( 2 \cdot 3 )\cdot2^{2}\cdot2^{2} \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(2\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(9\) (exact)

Modular invariants

Modular form 86190.2.a.bm

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)
 

\( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + 4q^{7} - q^{8} + q^{9} - q^{10} + q^{12} - 4q^{14} + q^{15} + q^{16} + q^{17} - q^{18} + 4q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 557383680
\( \Gamma_0(N) \)-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])
 

\( L(E,1) \) ≈ \( 4.01516866138 \)

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\((j)_{-}\)
\(2\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2
\(3\) \(2\) \( I_{2} \) Split multiplicative -1 1 2 2
\(5\) \(6\) \( I_{6} \) Split multiplicative -1 1 6 6
\(13\) \(4\) \( I_3^{*} \) Additive 1 2 9 3
\(17\) \(4\) \( I_{4} \) Split multiplicative -1 1 4 4

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 X13f.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 3 \\ 0 & 1 \end{array}\right)$ and has index 12.

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
\(2\) B
\(3\) B

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]
 

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$ 2 3 5 13 17
Reduction type nonsplit split split add split
$\lambda$-invariant(s) 4 7 1 - 3
$\mu$-invariant(s) 2 1 0 - 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

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 \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 86190bm consists of 8 curves linked by isogenies of degrees dividing 12.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 \(\Q(\sqrt{-39}) \) \(\Z/6\Z\) Not in database
\(\Q(\sqrt{-1}) \) \(\Z/4\Z\) Not in database
\(\Q(\sqrt{13}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
\(\Q(\sqrt{-13}) \) \(\Z/4\Z\) Not in database
4 \(\Q(i, \sqrt{39})\) \(\Z/12\Z\) Not in database
\(\Q(\sqrt{3}, \sqrt{-13})\) \(\Z/12\Z\) Not in database
\(\Q(\sqrt{-3}, \sqrt{13})\) \(\Z/2\Z \times \Z/6\Z\) Not in database
\(\Q(i, \sqrt{13})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
4.2.31636800.1 \(\Z/2\Z \times \Z/4\Z\) Not in database
6 \( x^{6} - 3 x^{5} - 6 x^{4} - 1207 x^{3} + 1854 x^{2} - 12879 x + 380637 \) \(\Z/2\Z \times \Z/6\Z\) 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.