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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])

gp: E = ellinit([0, 0, 1, -7, 6])

magma: E := EllipticCurve([0, 0, 1, -7, 6]);

$$y^2+y=x^3-7x+6$$ ## Mordell-Weil group structure

$\Z^3$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(1, 0\right)$$ $$\left(2, 0\right)$$ $$\left(0, 2\right)$$ $\hat{h}(P)$ ≈ $0.66820516565192793503314205089$ $0.76704335533154620579545064655$ $0.99090633315308797388259855289$

## 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)  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: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.56139014229398666466212500182\dots$ Stable Faltings height: $-0.56139014229398666466212500182\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $3$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.41714355875838396981711954462\dots$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $4.1516879830869330498841756835\dots$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $1$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $1$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (rounded) sage: r = E.rank(); sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: ar = ellanalyticrank(E); gp: ar/factorial(ar)  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); Special value: $L^{(3)}(E,1)/3!$ ≈ $1.7318499001193006897919750850614576885$

## Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - 2q^{2} - 3q^{3} + 2q^{4} - 4q^{5} + 6q^{6} - 4q^{7} + 6q^{9} + 8q^{10} - 6q^{11} - 6q^{12} - 4q^{13} + 8q^{14} + 12q^{15} - 4q^{16} - 4q^{17} - 12q^{18} - 7q^{19} + O(q^{20})$$ sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1984 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is semistable. There is only one prime of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)

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

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 $\ell$-adic Galois representation has maximal image $\GL(2,\Z_\ell)$ for all primes $\ell$.

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

Note: $p$-adic regulator data only exists for primes $p\ge 5$ 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 less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 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 $8$ Deg 8 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.

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).