Properties

Label 5077.a1
Conductor \(5077\)
Discriminant \(5077\)
j-invariant \( \frac{37933056}{5077} \)
CM no
Rank \(3\)
Torsion Structure \(\mathrm{Trivial}\)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

Historical Information about the Gauss elliptic curve

This elliptic curve has smallest conductor amongst those of rank 3.

In 1985, Buhler, Gross and Zagier used the celebrated Gross-Zagier Theorem on heights of Heegner points (see Gross, Benedict H.; Zagier, Don B. (1986), "Heegner points and derivatives of L-series", Inventiones Mathematicae 84 (2): 225–320, doi:10.1007/BF01388809) to prove that the L-function of this curve has a zero of order 3 at its critical point $s=1$, thus establishing the first part of the Birch and Swinnerton-Dyer conjecture for this curve (see Math. Comp. 44 (1985), 473-481). This was the first time that BSD had been established for any elliptic curve of rank $3$. To this day, it is not possible even in principle to establish BSD for any curve of rank $4$ or greater, since there is no known method for rigourously establishing the value of the analytic rank when it is greater than $3$.

Via Goldfeld's method, which required the use of an L-function of analytic rank at least $3$, this elliptic curve also found an application in the context of obtaining explicit lower bounds for the class numbers of imaginary quadratic fields. This solved Gauss's Class Number Problem first posed by Gauss in 1801 is his book Disquisitiones Arithmeticae (Section V, Articles 303 and 304).

Minimal Weierstrass equation

magma: E := EllipticCurve([0, 0, 1, -7, 6]); // or
magma: E := EllipticCurve("5077a1");
sage: E = EllipticCurve([0, 0, 1, -7, 6]) # or
sage: E = EllipticCurve("5077a1")
gp: E = ellinit([0, 0, 1, -7, 6]) \\ or
gp: E = ellinit("5077a1")

\( y^2 + y = x^{3} - 7 x + 6 \)

Mordell-Weil group structure

\(\Z^3\)

Infinite order Mordell-Weil generators and heights

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(-2, 3\right) \)\( \left(-1, 3\right) \)\( \left(0, 2\right) \)
\(\hat{h}(P)\) ≈  1.368572505351.205081104190.990906333153

Integral points

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

\( \left(-3, 0\right) \), \( \left(-2, 3\right) \), \( \left(-1, 3\right) \), \( \left(0, 2\right) \), \( \left(1, 0\right) \), \( \left(2, 0\right) \), \( \left(3, 3\right) \), \( \left(4, 6\right) \), \( \left(8, 21\right) \), \( \left(11, 35\right) \), \( \left(14, 51\right) \), \( \left(21, 95\right) \), \( \left(37, 224\right) \), \( \left(52, 374\right) \), \( \left(93, 896\right) \), \( \left(342, 6324\right) \), \( \left(406, 8180\right) \), \( \left(816, 23309\right) \)

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

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
\( N \)  =  \( 5077 \)  =  \(5077\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
\(\Delta\)  =  \(5077 \)  =  \(5077 \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
\(j \)  =  \( \frac{37933056}{5077} \)  =  \(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 5077^{-1}\)
\( \text{End} (E) \)  =  \(\Z\)   (no Complex Multiplication)
\( \text{ST} (E) \)  =  $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
sage: E.rank()
\( r \)  =  \(3\)
magma: Regulator(E);
sage: E.regulator()
\( \text{Reg} \)  ≈  \(0.417143558758\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
\( \Omega \)  ≈  \(4.15168798309\)
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\(_{\text{an}} \)  ≈   \(1\) (rounded)

Modular invariants

Modular form 5077.2.1.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)

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

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

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
1984 : 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])

\( L^{(3)}(E,1)/3! \) ≈ \( 1.73184990012 \)

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\((j)_{-}\)
\(5077\) \(1\) \( I_{1} \) 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

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

Note: \(p\)-adic data only exists for primes \(p\ge5\) of good ordinary reduction.

Isogenies

This curve has no rational isogenies. Its isogeny class 5077.a consists of this curve only.