# Properties

 Label 32244.a1 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()

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

## 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 32244.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 3.1.2687.1 $$\Z/2\Z$$ 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.