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Properties

Label 439569b1
Conductor $439569$
Discriminant $-1.365\times 10^{15}$
j-invariant \( -\frac{53248}{459} \)
CM no
Rank $2$
Torsion structure trivial

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Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 1, -11271, -1836234])
 
gp: E = ellinit([0, 0, 1, -11271, -1836234])
 
magma: E := EllipticCurve([0, 0, 1, -11271, -1836234]);
 

\(y^2+y=x^3-11271x-1836234\)

Mordell-Weil group structure

\(\Z^2\)

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(272, 3901\right) \)\( \left(731, 19507\right) \)
\(\hat{h}(P)\) ≈  $1.1283652502609456381742228035$$1.2400907802707039471631402506$

Integral points

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

\( \left(153, 144\right) \), \( \left(153, -145\right) \), \( \left(185, 1552\right) \), \( \left(185, -1553\right) \), \( \left(272, 3901\right) \), \( \left(272, -3902\right) \), \( \left(731, 19507\right) \), \( \left(731, -19508\right) \), \( \left(1598, 63724\right) \), \( \left(1598, -63725\right) \), \( \left(6887, 571468\right) \), \( \left(6887, -571469\right) \), \( \left(41310, 8396172\right) \), \( \left(41310, -8396173\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 439569 \)  =  \(3^{2} \cdot 13^{2} \cdot 17^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-1364961641011371 \)  =  \(-1 \cdot 3^{9} \cdot 13^{2} \cdot 17^{7} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{53248}{459} \)  =  \(-1 \cdot 2^{12} \cdot 3^{-3} \cdot 13 \cdot 17^{-1}\)
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: \(2\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(1.3553760699091694624156646148\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.20328607256664537474980666529\)
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 439569.2.a.b

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);
 

\( q - 2q^{2} + 2q^{4} - 2q^{5} - 4q^{7} + 4q^{10} - 3q^{11} + 8q^{14} - 4q^{16} - 7q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 3483648
\( \Gamma_0(N) \)-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);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 4.4084652496424006057189372491885186816 \)

Local data

This elliptic curve is not semistable. There are 3 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\((j)_{-}\)
\(3\) \(4\) \(I_3^{*}\) Additive -1 2 9 3
\(13\) \(1\) \(II\) Additive 1 2 2 0
\(17\) \(4\) \(I_1^{*}\) Additive 1 2 7 1

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 \( p \) .

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 no rational isogenies. Its isogeny class 439569b consists of this curve only.