# Properties

 Label 32244c1 Conductor 32244 Discriminant -386928 j-invariant $$-\frac{192914176}{24183}$$ CM no Rank 3 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, -1, 0, -30, 81]); // or
magma: E := EllipticCurve("32244c1");
sage: E = EllipticCurve([0, -1, 0, -30, 81]) # or
sage: E = EllipticCurve("32244c1")
gp: E = ellinit([0, -1, 0, -30, 81]) \\ or
gp: E = ellinit("32244c1")

$$y^2 = x^{3} - x^{2} - 30 x + 81$$

## Mordell-Weil group structure

$$\Z^3$$

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);
sage: E.gens()

 $$P$$ = $$\left(6, 9\right)$$ $$\left(-5, 9\right)$$ $$\left(-4, 11\right)$$ $$\hat{h}(P)$$ ≈ 0.803413421999 2.20431026124 1.67661898541

## Integral points

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

$$\left(-6, 3\right)$$, $$\left(-5, 9\right)$$, $$\left(-4, 11\right)$$, $$\left(0, 9\right)$$, $$\left(2, 5\right)$$, $$\left(3, 3\right)$$, $$\left(4, 3\right)$$, $$\left(6, 9\right)$$, $$\left(8, 17\right)$$, $$\left(11, 31\right)$$, $$\left(14, 47\right)$$, $$\left(27, 135\right)$$, $$\left(30, 159\right)$$, $$\left(48, 327\right)$$, $$\left(60, 459\right)$$, $$\left(156, 1941\right)$$, $$\left(238, 3663\right)$$, $$\left(410, 8291\right)$$, $$\left(652, 16635\right)$$, $$\left(9548, 932923\right)$$

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$32244$$ = $$2^{2} \cdot 3 \cdot 2687$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$-386928$$ = $$-1 \cdot 2^{4} \cdot 3^{2} \cdot 2687$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$-\frac{192914176}{24183}$$ = $$-1 \cdot 2^{8} \cdot 3^{-2} \cdot 7^{3} \cdot 13^{3} \cdot 2687^{-1}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E); sage: E.rank() Rank: $$3$$ magma: Regulator(E); sage: E.regulator() Regulator: $$0.394959406305$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$2.9164743935$$ 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: $$6$$  = $$3\cdot2\cdot1$$ 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 32244.2.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)

$$q - q^{3} - 4q^{5} - 5q^{7} + q^{9} - 4q^{11} - 7q^{13} + 4q^{15} - 6q^{17} - 3q^{19} + O(q^{20})$$

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 13344 $$\Gamma_0(N)$$-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])

$$L^{(3)}(E,1)/3!$$ ≈ $$6.91133396977$$

## 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$$(j)_{-}$$
$$2$$ $$3$$ $$IV$$ Additive -1 2 4 0
$$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$2687$$ $$1$$ $$I_{1}$$ 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]

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 2687 add nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary split - 3 3 3 5 3 3 5 3 3 3 3 3 3 3 ? - 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.

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

## Isogenies

This curve has no rational isogenies. Its isogeny class 32244c 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 3.1.2687.1 $$\Z/2\Z$$ Not in database
6 6.0.19400056703.1 $$\Z/2\Z \times \Z/2\Z$$ 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.