# Properties

 Label 3380f2 Conductor $3380$ Discriminant $11424400$ j-invariant $$\frac{151635187115776}{25}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -85570, -9663107])

gp: E = ellinit([0, 1, 0, -85570, -9663107])

magma: E := EllipticCurve([0, 1, 0, -85570, -9663107]);

$$y^2=x^3+x^2-85570x-9663107$$

## Mordell-Weil group structure

$\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(-\frac{1523}{9}, \frac{1}{27}\right)$$ (-1523/9, 1/27) $\hat{h}(P)$ ≈ $2.4529663727190638379026455326$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$3380$$ = $2^{2} \cdot 5 \cdot 13^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $11424400$ = $2^{4} \cdot 5^{2} \cdot 13^{4}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{151635187115776}{25}$$ = $2^{8} \cdot 5^{-2} \cdot 7^{6} \cdot 13^{2} \cdot 31^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.1970663742342244994396580979\dots$ Stable Faltings height: $0.11103419489373048428275157689\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $2.4529663727190638379026455326\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.27913786261677980773640465755\dots$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $6$  = $1\cdot2\cdot3$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $1$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (exact) sage: r = E.rank(); sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: ar = ellanalyticrank(E); gp: ar[2]/factorial(ar[1])  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); Special value: $L'(E,1)$ ≈ $4.1082947421098084046191349538$

## Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

magma: ModularForm(E);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 5184 $\Gamma_0(N)$-optimal: no Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 3 primes of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $1$ $IV$ Additive -1 2 4 0
$5$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$13$ $3$ $IV$ Additive 1 2 4 0

## Galois representations

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2Cn 2.2.0.1
$3$ 3B.1.2 3.8.0.2

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

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

## 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 ord split ord ord add ord ord ord ord ord ord ord ord ss - 3 2 1 1 - 1 1 1 1 1 1 1 1 1,1 - 1 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 non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 3380f consists of 2 curves linked by isogenies of degree 3.

## Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-3})$$ $$\Z/3\Z$$ Not in database $3$ 3.3.169.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $3$ 3.1.456300.2 $$\Z/3\Z$$ Not in database $6$ 6.0.624629070000.2 $$\Z/3\Z \times \Z/3\Z$$ Not in database $6$ 6.0.771147.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database $9$ 9.3.95006081547000000.15 $$\Z/2\Z \times \Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/4\Z$$ Not in database $18$ 18.0.284258910904006648725908395200000000.1 $$\Z/9\Z$$ Not in database $18$ 18.0.243706199334710775656643000000000000.10 $$\Z/6\Z \times \Z/6\Z$$ Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.