Properties

Label 28322.j2
Conductor 28322
Discriminant -5469084072142256776
j-invariant \( \frac{63905303}{39304} \)
CM no
Rank 1
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, 431616, -27166936]) # or
 
sage: E = EllipticCurve("28322b2")
 
gp: E = ellinit([1, 1, 0, 431616, -27166936]) \\ or
 
gp: E = ellinit("28322b2")
 
magma: E := EllipticCurve([1, 1, 0, 431616, -27166936]); // or
 
magma: E := EllipticCurve("28322b2");
 

\( y^2 + x y = x^{3} + x^{2} + 431616 x - 27166936 \)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(\frac{45197117}{163216}, \frac{693935125129}{65939264}\right) \)
\(\hat{h}(P)\) ≈  15.852804745872128

Integral points

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

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 28322 \)  =  \(2 \cdot 7^{2} \cdot 17^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-5469084072142256776 \)  =  \(-1 \cdot 2^{3} \cdot 7^{8} \cdot 17^{9} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{63905303}{39304} \)  =  \(2^{-3} \cdot 7 \cdot 11^{3} \cdot 17^{-3} \cdot 19^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Rank: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(15.8528047459\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.139381744736\)
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: \( 2 \)  = \( 1\cdot1\cdot2 \)
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: \(1\) (exact)

Modular invariants

Modular form 28322.2.a.j

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);
 

\( q - q^{2} + 2q^{3} + q^{4} - 3q^{5} - 2q^{6} - q^{8} + q^{9} + 3q^{10} + 2q^{12} + 2q^{13} - 6q^{15} + q^{16} - q^{18} - 7q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 725760
\( \Gamma_0(N) \)-optimal: no
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);
 

\( L'(E,1) \) ≈ \( 4.41918316888 \)

Local data

This elliptic curve is not semistable.

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\((j)_{-}\)
\(2\) \(1\) \( I_{3} \) Non-split multiplicative 1 1 3 3
\(7\) \(1\) \( IV^{*} \) Additive 1 2 8 0
\(17\) \(2\) \( I_3^{*} \) Additive 1 2 9 3

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

$p$-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 nonsplit ordinary ordinary add ss ordinary add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss
$\lambda$-invariant(s) 6 1 1 - 1,1 1 - 1 1 1 1 1 3 1 1,1
$\mu$-invariant(s) 0 1 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 28322.j 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 \(\Q(\sqrt{-51}) \) \(\Z/3\Z\) Not in database
3 3.1.6664.1 \(\Z/2\Z\) Not in database
6 6.0.6039609856.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
6.2.8599366377.7 \(\Z/3\Z\) Not in database
6.0.20383683264.8 \(\Z/6\Z\) 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.