Properties

Label 21443.a1
Conductor 21443
Discriminant -21443
j-invariant \( -\frac{13997521}{21443} \)
CM no
Rank 3
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, -5, 6]) # or
 
sage: E = EllipticCurve("21443a1")
 
gp: E = ellinit([1, 1, 1, -5, 6]) \\ or
 
gp: E = ellinit("21443a1")
 
magma: E := EllipticCurve([1, 1, 1, -5, 6]); // or
 
magma: E := EllipticCurve("21443a1");
 

\( y^2 + x y + y = x^{3} + x^{2} - 5 x + 6 \)

Mordell-Weil group structure

\(\Z^3\)

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(-3, 3\right) \)\( \left(-2, 4\right) \)\( \left(0, 2\right) \)
\(\hat{h}(P)\) ≈  1.44415127309442571.3355737463786821.0903075273991985

Integral points

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

\( \left(-3, 3\right) \), \( \left(-3, -1\right) \), \( \left(-2, 4\right) \), \( \left(-2, -3\right) \), \( \left(0, 2\right) \), \( \left(0, -3\right) \), \( \left(1, 1\right) \), \( \left(1, -3\right) \), \( \left(4, 6\right) \), \( \left(4, -11\right) \), \( \left(6, 12\right) \), \( \left(6, -19\right) \), \( \left(18, 69\right) \), \( \left(18, -88\right) \), \( \left(34, 184\right) \), \( \left(34, -219\right) \), \( \left(253, 3907\right) \), \( \left(253, -4161\right) \), \( \left(39862, 7938829\right) \), \( \left(39862, -7978692\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 21443 \)  =  \(41 \cdot 523\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-21443 \)  =  \(-1 \cdot 41 \cdot 523 \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{13997521}{21443} \)  =  \(-1 \cdot 41^{-1} \cdot 241^{3} \cdot 523^{-1}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Rank: \(3\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(1.24510112449\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(3.43484841508\)
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: \( 1 \)  = \( 1\cdot1 \)
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 21443.2.a.a

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);
 

\( q - q^{2} - q^{3} - q^{4} - 4q^{5} + q^{6} - 2q^{7} + 3q^{8} - 2q^{9} + 4q^{10} - 6q^{11} + q^{12} - 4q^{13} + 2q^{14} + 4q^{15} - q^{16} + 4q^{17} + 2q^{18} - 6q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 2912
\( \Gamma_0(N) \)-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);
 

\( L^{(3)}(E,1)/3! \) ≈ \( 4.27673362407 \)

Local data

This elliptic curve is 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\((j)_{-}\)
\(41\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1
\(523\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 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(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 47 523
Reduction type ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary nonsplit ordinary ordinary nonsplit
$\lambda$-invariant(s) 3 3 3 9 5 3 3 3 3 3 3 3 3 3 3 3
$\mu$-invariant(s) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Isogenies

This curve has no rational isogenies. Its isogeny class 21443.a 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 3.1.21443.1 \(\Z/2\Z\) Not in database
6 6.0.9859539625307.1 \(\Z/2\Z \times \Z/2\Z\) 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.