Properties

Label 88.a1
Conductor 88
Discriminant -2816
j-invariant \( -\frac{27648}{11} \)
CM no
Rank 1
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([0, 0, 0, -4, 4]); // or
magma: E := EllipticCurve("88a1");
sage: E = EllipticCurve([0, 0, 0, -4, 4]) # or
sage: E = EllipticCurve("88a1")
gp: E = ellinit([0, 0, 0, -4, 4]) \\ or
gp: E = ellinit("88a1")

\( y^2 = x^{3} - 4 x + 4 \)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(2, -2\right) \)
\(\hat{h}(P)\) ≈  0.0402643643369

Integral points

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

\( \left(-2, 2\right) \), \( \left(0, 2\right) \), \( \left(1, 1\right) \), \( \left(2, 2\right) \), \( \left(6, 14\right) \), \( \left(8, 22\right) \), \( \left(310, 5458\right) \)

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

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
Conductor: \( 88 \)  =  \(2^{3} \cdot 11\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
Discriminant: \(-2816 \)  =  \(-1 \cdot 2^{8} \cdot 11 \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( -\frac{27648}{11} \)  =  \(-1 \cdot 2^{10} \cdot 3^{3} \cdot 11^{-1}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
sage: E.rank()
Rank: \(1\)
magma: Regulator(E);
sage: E.regulator()
Regulator: \(0.0402643643369\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
Real period: \(4.25252953315\)
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: \( 4 \)  = \( 2^{2}\cdot1 \)
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: \(1\) (exact)

Modular invariants

Modular form 88.2.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)

\( q - 3q^{3} - 3q^{5} - 2q^{7} + 6q^{9} - q^{11} + 9q^{15} - 6q^{17} + 4q^{19} + O(q^{20}) \)

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

magma: ModularDegree(E);
sage: E.modular_degree()
Modular degree: 8
\( \Gamma_0(N) \)-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])

\( L'(E,1) \) ≈ \( 0.684901593904 \)

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\((j)_{-}\)
\(2\) \(4\) \( I_1^{*} \) Additive 1 3 8 0
\(11\) \(1\) \( I_{1} \) 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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type add ss ordinary ordinary nonsplit ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) - 1,1 1 1 1 1,1 1 1 1 5 1 1 3 1 1
$\mu$-invariant(s) - 0,0 0 0 0 0,0 0 0 0 0 0 0 0 0 0

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has no rational isogenies. Its isogeny class 88.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 3.1.44.1 \(\Z/2\Z\) Not in database
6 6.0.21296.1 \(\Z/2\Z \times \Z/2\Z\) 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.

Additional information

$E$ parametrizes triangles $ABC$ with rational sides $a,b,c$ for which the altitude from $A$, angle bisector to $B$, and median from $C$ are concurrent. Equivalently, $c (a^2 + b^2 - c^2) = a (a^2 + c^2 - b^2)$ [proof by standard triangle geometry, including "Ceva's theorem"]. This elliptic curve is put in standard Weierstrass form by taking $c = ((2/x) - 1) a$, when $y^2 = x^3 - 4x + 4$. The generator $(x,y)=(2,2)$ of the Mordell-Weil group corresponds to an equilateral triangle. The higher multiples that yield positive $(a:b:c)$ are the 7th, 10th, and 12th, with x-coordinates $10/9$, $88/49$, $206/961$, and triangles $(a:b:c) = (15:13:12), (308:277:35), (3193:26447:26598).$ This has been independently observed many times, going back at least to 1939; see [Albime triangles and Guy's favourite elliptic curve] (in Expo. Math. 2015) by Erika Bakker, Jasbir S. Chahal, and Jaap Top.

The title of the Bakker-Chahal-Top paper (and of this knowl) come from Richard Guy's paper "My Favorite Elliptic Curve: A Tale of Two Types of Triangles", Amer. Math. Monthly 102 #9 (Nov. 1995), 771-781, where the same curve also arises in two other contexts; notably, it parametrizes pairs $(R,T)$ of a rectangle $R$ and an isosceles triangle $R$, both with rational sides, and with the same perimeter and area. (For example, $R$ can be a $2 \times 6$ rectangle of perimeter $16$ and area $12$, same as a 5-5-6 triangle $T$ made from two copies of the Pythagorean 3-4-5.)