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

## 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()

 $$P$$ = $$\left(-2, 3\right)$$ $$\left(-1, 3\right)$$ $$\left(0, 2\right)$$ $$\hat{h}(P)$$ ≈ 1.36857250535 1.20508110419 0.990906333153

## 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) Conductor: $$5077$$ = $$5077$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$5077$$ = $$5077$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j 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

 magma: Rank(E); sage: E.rank() Rank: $$3$$ magma: Regulator(E); sage: E.regulator() Regulator: $$0.417143558758$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega Real period: $$4.15168798309$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$1$$  = $$1$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E) Torsion order: $$1$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

#### Modular form5077.2.a.a

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

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 1984 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

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/factorial(ar)

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

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(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.

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

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 5077 ss ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ss ordinary ordinary nonsplit 4,3 3,3 3 3 3 3 3 3 3 3 3 3,3 3,3 3 3 ? 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.

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).