# Properties

 Label 3380.i2 Conductor $3380$ Discriminant $67600$ j-invariant $$\frac{1141504}{25}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -30, 53])

gp: E = ellinit([0, 1, 0, -30, 53])

magma: E := EllipticCurve([0, 1, 0, -30, 53]);

$$y^2=x^3+x^2-30x+53$$

## Mordell-Weil group structure

$\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(1, 5\right)$$ (1, 5) $\hat{h}(P)$ ≈ $0.59841102432194606251689442053$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(1,\pm 5)$$

## 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: $67600$ = $2^{4} \cdot 5^{2} \cdot 13^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{1141504}{25}$$ = $2^{8} \cdot 5^{-2} \cdot 7^{3} \cdot 13$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.28511268353052002979978405550\dots$ Stable Faltings height: $-0.94365330329409125561444266958\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.59841102432194606251689442053\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $3.4730211936239782770529422605\dots$ 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: $2$  = $1\cdot2\cdot1$ 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: $L'(E,1)$ ≈ $4.1565883399367052200999429455$

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 288 $\Gamma_0(N)$-optimal: yes 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
$13$ $1$ $II$ Additive 1 2 2 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 3.4.0.1

## $p$-adic regulators

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

Note: $p$-adic regulator data only exists for primes $p\ge 5$ 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 ord split ord ord add ord ord ord ord ord ord ord ord ss - 3 2 1 1 - 1 1 1 1 3 1 1 1 1,1 - 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 non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 3380.i 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 $K$ $2$ $$\Q(\sqrt{13})$$ $$\Z/3\Z$$ Not in database $3$ 3.3.169.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.0.100249110000.3 $$\Z/3\Z$$ Not in database $6$ $$\Q(\zeta_{13})^+$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/4\Z$$ Not in database $12$ Deg 12 $$\Z/3\Z \times \Z/3\Z$$ Not in database $18$ 18.6.31728741922273159947712276800000000.1 $$\Z/9\Z$$ Not in database $18$ 18.0.1007491932196348370031000000000000.5 $$\Z/2\Z \times \Z/6\Z$$ 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.