# Properties

 Label 234446.a1 Conductor 234446 Discriminant 468892 j-invariant $$\frac{54915331401}{468892}$$ CM no Rank 4 Torsion Structure $$\mathrm{Trivial}$$

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This elliptic curve has smallest conductor amongst those of rank 4.

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 0, -79, 289]); // or
magma: E := EllipticCurve("234446a1");
sage: E = EllipticCurve([1, -1, 0, -79, 289]) # or
sage: E = EllipticCurve("234446a1")
gp: E = ellinit([1, -1, 0, -79, 289]) \\ or
gp: E = ellinit("234446a1")

$$y^2 + x y = x^{3} - x^{2} - 79 x + 289$$

## Mordell-Weil group structure

$$\Z^4$$

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);
sage: E.gens()

 $$P$$ = $$\left(4, 3\right)$$ $$\left(5, -2\right)$$ $$\left(6, -1\right)$$ $$\left(8, 7\right)$$ $$\hat{h}(P)$$ ≈ 1.17647633592 1.20262600414 0.9837083405 1.51275519856

## Integral points

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

$$\left(-10, 7\right)$$, $$\left(-9, 19\right)$$, $$\left(-8, 23\right)$$, $$\left(-7, 25\right)$$, $$\left(-4, 25\right)$$, $$\left(0, 17\right)$$, $$\left(1, 14\right)$$, $$\left(3, 7\right)$$, $$\left(4, 3\right)$$, $$\left(5, -2\right)$$, $$\left(6, -1\right)$$, $$\left(7, 3\right)$$, $$\left(8, 7\right)$$, $$\left(12, 25\right)$$, $$\left(13, 30\right)$$, $$\left(22, 83\right)$$, $$\left(27, 118\right)$$, $$\left(29, 133\right)$$, $$\left(38, 207\right)$$, $$\left(60, 427\right)$$, $$\left(70, 543\right)$$, $$\left(91, 815\right)$$, $$\left(123, 1295\right)$$, $$\left(129, 1393\right)$$, $$\left(176, 2239\right)$$, $$\left(292, 4835\right)$$, $$\left(992, 30735\right)$$, $$\left(1140, 37907\right)$$, $$\left(1656, 66545\right)$$, $$\left(4532, 302803\right)$$, $$\left(10583, 1083382\right)$$, $$\left(19405, 2693397\right)$$

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$234446$$ = $$2 \cdot 117223$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$468892$$ = $$2^{2} \cdot 117223$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{54915331401}{468892}$$ = $$2^{-2} \cdot 3^{3} \cdot 7^{3} \cdot 181^{3} \cdot 117223^{-1}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E); sage: E.rank() Rank: $$4$$ magma: Regulator(E); sage: E.regulator() Regulator: $$1.50434488828$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$2.97267184726$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$2$$  = $$2\cdot1$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] Torsion order: $$1$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (rounded)

## Modular invariants

#### Modular form 234446.2.a.a

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

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

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

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 334976 $$\Gamma_0(N)$$-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[2]/factorial(ar[1])

$$L^{(4)}(E,1)/4!$$ ≈ $$8.9438473959$$

## 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$117223$$ $$1$$ $$I_{1}$$ 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]

No $$p$$-adic data exists for this curve.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has no rational isogenies. Its isogeny class 234446.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 3.3.468892.1 $$\Z/2\Z$$ Not in database
6 $$x^{6}$$ $$\mathstrut -\mathstrut 386 x^{4}$$ $$\mathstrut +\mathstrut 37249 x^{2}$$ $$\mathstrut -\mathstrut 468892$$ $$\Z/2\Z \times \Z/2\Z$$ 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

This is the first case where the minimal conductor $N$ of an elliptic curve of given rank $r$ is not prime. (For $r=0,1,2,3$ the minima are $11,37,389,5077$.) It is also the first case where the record is not attained by a curve of discriminant $\pm N$. [NB one of the three curves of conductor $11$, namely [11.a2] (a.k.a. the modular curve $X_0(11)$, has discriminant $-11^5$, but the other two have discriminant $-11$.] Still, the discriminant $468892$ of this curve is expected to have the smallest absolute value among all elliptic curves of rank at least $4$; the second-smallest known is the prime $501029$, for the curve $[0, 1, 1, -72, 210]$.