Properties

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

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This elliptic curve has smallest conductor amongst elliptic curves over $\Q$ of rank 3.

Minimal Weierstrass equation

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

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

Mordell-Weil group structure

\(\Z^3\)

Infinite order Mordell-Weil generators and heights

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

\(P\) =  \( \left(-2, 3\right) \)\( \left(-1, 3\right) \)\( \left(0, 2\right) \)
\(\hat{h}(P)\) ≈  1.36857250535393021.20508110418585220.990906333153088

Integral points

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

\( \left(-3, 0\right) \), \( \left(-3, -1\right) \), \( \left(-2, 3\right) \), \( \left(-2, -4\right) \), \( \left(-1, 3\right) \), \( \left(-1, -4\right) \), \( \left(0, 2\right) \), \( \left(0, -3\right) \), \( \left(1, 0\right) \), \( \left(1, -1\right) \), \( \left(2, 0\right) \), \( \left(2, -1\right) \), \( \left(3, 3\right) \), \( \left(3, -4\right) \), \( \left(4, 6\right) \), \( \left(4, -7\right) \), \( \left(8, 21\right) \), \( \left(8, -22\right) \), \( \left(11, 35\right) \), \( \left(11, -36\right) \), \( \left(14, 51\right) \), \( \left(14, -52\right) \), \( \left(21, 95\right) \), \( \left(21, -96\right) \), \( \left(37, 224\right) \), \( \left(37, -225\right) \), \( \left(52, 374\right) \), \( \left(52, -375\right) \), \( \left(93, 896\right) \), \( \left(93, -897\right) \), \( \left(342, 6324\right) \), \( \left(342, -6325\right) \), \( \left(406, 8180\right) \), \( \left(406, -8181\right) \), \( \left(816, 23309\right) \), \( \left(816, -23310\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 5077 \)  =  \(5077\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(5077 \)  =  \(5077 \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{37933056}{5077} \)  =  \(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 5077^{-1}\)
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: \(0.417143558758\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(4.15168798309\)
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: \( 1 \)  = \( 1 \)
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 5077.2.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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 1984
\( \Gamma_0(N) \)-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);
 

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

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\((j)_{-}\)
\(5077\) \(1\) \( I_{1} \) 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 5077
Reduction type ss ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ss ordinary ordinary nonsplit
$\lambda$-invariant(s) 4,3 3,3 3 3 3 3 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 0,0 0 0 ?

An entry ? indicates that the invariants have not yet been computed.

Isogenies

This curve has no rational isogenies. Its isogeny class 5077.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 3.3.20308.1 \(\Z/2\Z\) Not in database
6 6.6.2093830264528.1 \(\Z/2\Z \times \Z/2\Z\) 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.

Additional information

Historical Information about the Gauss elliptic curve

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, [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: [10.1090/S0025-5718-1985-0777279-X]). 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).