# Properties

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

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## 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 form88.2.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)

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

 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

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 add ss ordinary ordinary nonsplit ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary - 1,1 1 1 1 1,1 1 1 1 5 1 1 3 1 1 - 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 88a 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.

$E$ 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.)