Properties

Label 86190.by1
Conductor 86190
Discriminant 72741988051200000
j-invariant \( \frac{4752182606640001}{2546899200000} \)
CM no
Rank 2
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 1, -107065, -3709945]); // or
 
magma: E := EllipticCurve("86190cd1");
 
sage: E = EllipticCurve([1, 1, 1, -107065, -3709945]) # or
 
sage: E = EllipticCurve("86190cd1")
 
gp: E = ellinit([1, 1, 1, -107065, -3709945]) \\ or
 
gp: E = ellinit("86190cd1")
 

\( y^2 + x y + y = x^{3} + x^{2} - 107065 x - 3709945 \)

Mordell-Weil group structure

\(\Z^2\)

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(-307, 738\right) \)\( \left(-47, -1082\right) \)
\(\hat{h}(P)\) ≈  1.318289517780.271641984534

Integral points

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

\( \left(-307, 738\right) \), \( \left(-307, -432\right) \), \( \left(-285, 2080\right) \), \( \left(-285, -1796\right) \), \( \left(-217, 3168\right) \), \( \left(-217, -2952\right) \), \( \left(-177, 3208\right) \), \( \left(-177, -3032\right) \), \( \left(-137, 2968\right) \), \( \left(-137, -2832\right) \), \( \left(-125, 2844\right) \), \( \left(-125, -2720\right) \), \( \left(-47, 1128\right) \), \( \left(-47, -1082\right) \), \( \left(-37, 468\right) \), \( \left(-37, -432\right) \), \( \left(343, 88\right) \), \( \left(343, -432\right) \), \( \left(395, 3780\right) \), \( \left(395, -4176\right) \), \( \left(463, 6568\right) \), \( \left(463, -7032\right) \), \( \left(633, 13198\right) \), \( \left(633, -13832\right) \), \( \left(773, 19008\right) \), \( \left(773, -19782\right) \), \( \left(863, 22968\right) \), \( \left(863, -23832\right) \), \( \left(1279, 43560\right) \), \( \left(1279, -44840\right) \), \( \left(1295, 44424\right) \), \( \left(1295, -45720\right) \), \( \left(2163, 98368\right) \), \( \left(2163, -100532\right) \), \( \left(4373, 286218\right) \), \( \left(4373, -290592\right) \), \( \left(5903, 449928\right) \), \( \left(5903, -455832\right) \), \( \left(13183, 1506648\right) \), \( \left(13183, -1519832\right) \), \( \left(22963, 3467968\right) \), \( \left(22963, -3490932\right) \), \( \left(24263, 3766968\right) \), \( \left(24263, -3791232\right) \), \( \left(224463, 106232968\right) \), \( \left(224463, -106457432\right) \), \( \left(31128245, 173657298474\right) \), \( \left(31128245, -173688426720\right) \)

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: \(72741988051200000 \)  =  \(2^{11} \cdot 3^{4} \cdot 5^{5} \cdot 13^{4} \cdot 17^{3} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( \frac{4752182606640001}{2546899200000} \)  =  \(2^{-11} \cdot 3^{-4} \cdot 5^{-5} \cdot 13^{2} \cdot 17^{-3} \cdot 47^{3} \cdot 647^{3}\)
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: \(0.0484733588972\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.280772965082\)
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: \( 990 \)  = \( 11\cdot2\cdot5\cdot3\cdot3 \)
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 86190.2.a.by

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)
 

\( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - 3q^{7} + q^{8} + q^{9} + q^{10} - 5q^{11} - q^{12} - 3q^{14} - q^{15} + q^{16} + q^{17} + q^{18} - 2q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 1140480
\( \Gamma_0(N) \)-optimal: yes
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])
 

\( L^{(2)}(E,1)/2! \) ≈ \( 13.473908618 \)

Local data

This elliptic curve is not semistable.

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\((j)_{-}\)
\(2\) \(11\) \( I_{11} \) Split multiplicative -1 1 11 11
\(3\) \(2\) \( I_{4} \) Non-split multiplicative 1 1 4 4
\(5\) \(5\) \( I_{5} \) Split multiplicative -1 1 5 5
\(13\) \(3\) \( IV \) Additive 1 2 4 0
\(17\) \(3\) \( I_{3} \) 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 \( p \) .

$p$-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 \(p\ge5\) of good ordinary reduction.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type split nonsplit split ordinary ordinary add split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss
$\lambda$-invariant(s) 6 4 3 2 2 - 3 4 2 2 2 2 2 2 2,2
$\mu$-invariant(s) 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 no rational isogenies. Its isogeny class 86190.by consists of this curve only.

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
3 3.3.114920.1 \(\Z/2\Z\) Not in database
6 6.6.8980492352000.1 \(\Z/2\Z \times \Z/2\Z\) 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.