Properties

Label 43.a1
Conductor \(43\)
Discriminant \(-43\)
j-invariant \( -\frac{4096}{43} \)
CM no
Rank \(1\)
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 1, 0, 0]); // or
magma: E := EllipticCurve("43a1");
sage: E = EllipticCurve([0, 1, 1, 0, 0]) # or
sage: E = EllipticCurve("43a1")
gp: E = ellinit([0, 1, 1, 0, 0]) \\ or
gp: E = ellinit("43a1")

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

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(0, 0\right) \)
\(\hat{h}(P)\) ≈  0.0628165070875

Integral points

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

\( \left(-1, 0\right) \), \( \left(0, 0\right) \), \( \left(1, 1\right) \), \( \left(2, 3\right) \), \( \left(21, 98\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
\( N \)  =  \( 43 \)  =  \(43\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
\(\Delta\)  =  \(-43 \)  =  \(-1 \cdot 43 \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
\(j \)  =  \( -\frac{4096}{43} \)  =  \(-1 \cdot 2^{12} \cdot 43^{-1}\)
\( \text{End} (E) \)  =  \(\Z\)   (no Complex Multiplication)
\( \text{ST} (E) \)  =  $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
sage: E.rank()
\( r \)  =  \(1\)
magma: Regulator(E);
sage: E.regulator()
\( \text{Reg} \)  ≈  \(0.0628165070875\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
\( \Omega \)  ≈  \(5.46868952997\)
magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
\( \prod_p c_p \)  =  \( 1 \)  = \( 1 \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
\( \#E_{\text{tor}} \)  = \(1\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Ш\(_{\text{an}} \)  =   \(1\) (exact)

Modular invariants

Modular form 43.2.1.a

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

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

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

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
2 : curve is \( \Gamma_0(N) \)-optimal

Special L-value attached to the curve

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])

\( L'(E,1) \) ≈ \( 0.343523974618 \)

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\((j)_{-}\)
\(43\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1

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 ss ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary nonsplit ordinary
$\lambda$-invariant(s) 2,1 1 1 1,3 1 1 1 1 1 1 1 1,1 1 1 1
$\mu$-invariant(s) 0,0 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 43.a consists of this curve only.