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()

\(P\) =  \( \left(4, 3\right) \)\( \left(5, -2\right) \)\( \left(6, -1\right) \)\( \left(8, 7\right) \)
\(\hat{h}(P)\) ≈  1.176476335921.202626004140.98370834051.51275519856

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]

\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.

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 $[0, 1, 1, -72, 210]$.