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

## Simplified equation

 $$y^2+y=x^3-7x+6$$ y^2+y=x^3-7x+6 (homogenize, simplify) $$y^2z+yz^2=x^3-7xz^2+6z^3$$ y^2z+yz^2=x^3-7xz^2+6z^3 (dehomogenize, simplify) $$y^2=x^3-112x+400$$ y^2=x^3-112x+400 (homogenize, minimize)

comment: Define the curve

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

gp: E = ellinit([0, 0, 1, -7, 6])

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

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

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z \oplus \Z \oplus \Z$$

magma: MordellWeilGroup(E);

### Infinite order Mordell-Weil generators and heights

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

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Integral points

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

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$5077$$ = $5077$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E);  oscar: conductor(E) Discriminant: $5077$ = $5077$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{37933056}{5077}$$ = $2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 5077^{-1}$ comment: j-invariant  sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E);  oscar: j_invariant(E) Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.56139014229398666466212500182\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-0.56139014229398666466212500182\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E)

## BSD invariants

 Analytic rank: $3$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $0.41714355875838396981711954462\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $4.1516879830869330498841756835\dots$ comment: Real Period  sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Tamagawa product: $1$ comment: Tamagawa numbers  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E);  oscar: tamagawa_numbers(E) Torsion order: $1$ comment: Torsion order  sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E));  oscar: prod(torsion_structure(E)) Analytic order of Ш: $1$ (rounded) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L^{(3)}(E,1)/3!$ ≈ $1.7318499001193006897919750851$ comment: Special L-value  r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: [r,L1r] = ellanalyticrank(E); L1r/r!  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

## BSD formula

$\displaystyle 1.731849900 \approx L^{(3)}(E,1)/3! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 4.151688 \cdot 0.417144 \cdot 1}{1^2} \approx 1.731849900$

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)

E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;

Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();

omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();

assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))

/* self-contained Magma code snippet for the BSD formula (checks rank, computes analyiic sha) */

E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;

sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);

reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);

assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);

## Modular invariants

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

comment: q-expansion of modular form

sage: E.q_eigenform(20)

\\ actual modular form, use for small N

[mf,F] = mffromell(E)

Ser(mfcoefs(mf,20),q)

\\ or just the series

Ser(ellan(E,20),q)*q

magma: ModularForm(E);

Modular degree: 1984
comment: Modular degree

sage: E.modular_degree()

gp: ellmoddegree(E)

magma: ModularDegree(E);

$\Gamma_0(N)$-optimal: yes
Manin constant: 1
comment: Manin constant

magma: ManinConstant(E);

## Local data

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

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

comment: Local data

sage: E.local_data()

gp: ellglobalred(E)

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]

## Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$.

comment: mod p Galois image

sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

gens = [[5079, 2, 5079, 3], [1, 2, 0, 1], [1, 0, 2, 1], [1, 1, 10153, 0], [10153, 2, 10152, 3]]

GL(2,Integers(10154)).subgroup(gens)

Gens := [[5079, 2, 5079, 3], [1, 2, 0, 1], [1, 0, 2, 1], [1, 1, 10153, 0], [10153, 2, 10152, 3]];

sub<GL(2,Integers(10154))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$10154 = 2 \cdot 5077$$, index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 5079 & 2 \\ 5079 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 10153 & 0 \end{array}\right),\left(\begin{array}{rr} 10153 & 2 \\ 10152 & 3 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

The torsion field $K:=\Q(E)$ is a degree-$1992802876951968$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10154\Z)$.

## Isogenies

gp: ellisomat(E)

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

## Twists

This elliptic curve is its own minimal quadratic twist.

## 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 $$\Z/2\Z \oplus \Z/2\Z$$ 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.

## 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 ord ord ord ord ord ord ord ord ord ss ss ord ord 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.

## $p$-adic regulators

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

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).