# Properties

 Label 1342.b3 Conductor 1342 Discriminant 22515023872 j-invariant $$\frac{300872095888141441}{22515023872}$$ CM no Rank 0 Torsion Structure $$\Z/{5}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 1, -13960, 629001]); // or
magma: E := EllipticCurve("1342c1");
sage: E = EllipticCurve([1, 1, 1, -13960, 629001]) # or
sage: E = EllipticCurve("1342c1")
gp: E = ellinit([1, 1, 1, -13960, 629001]) \\ or
gp: E = ellinit("1342c1")

$$y^2 + x y + y = x^{3} + x^{2} - 13960 x + 629001$$

## Mordell-Weil group structure

$$\Z/{5}\Z$$

## Torsion generators

magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp: elltors(E)

$$\left(69, -19\right)$$

## Integral points

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

$$\left(69, -19\right)$$, $$\left(101, 461\right)$$

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$1342$$ = $$2 \cdot 11 \cdot 61$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$22515023872$$ = $$2^{25} \cdot 11 \cdot 61$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{300872095888141441}{22515023872}$$ = $$2^{-25} \cdot 11^{-1} \cdot 61^{-1} \cdot 89^{3} \cdot 7529^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E); sage: E.rank() Rank: $$0$$ magma: Regulator(E); sage: E.regulator() Regulator: $$1$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$1.14697680979$$ 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: $$25$$  = $$5^{2}\cdot1\cdot1$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] Torsion order: $$5$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form1342.2.a.b

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)

$$q + q^{2} - q^{3} + q^{4} - 4q^{5} - q^{6} - 2q^{7} + q^{8} - 2q^{9} - 4q^{10} + q^{11} - q^{12} - q^{13} - 2q^{14} + 4q^{15} + q^{16} + 3q^{17} - 2q^{18} + O(q^{20})$$

#### Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
3200 . This curve is $$\Gamma_0(N)$$-optimal.

#### 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])

$$L(E,1)$$ ≈ $$1.14697680979$$

## 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$$(j)_{-}$$
$$2$$ $$25$$ $$I_{25}$$ Split multiplicative -1 1 25 25
$$11$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$61$$ $$1$$ $$I_{1}$$ 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 $$p$$ except those listed.

prime Image of Galois representation
$$5$$ B.1.1

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

All $$p$$-adic regulators are identically $$1$$ since the rank is $$0$$.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 11 61 split ordinary ordinary split split 10 0 2 1 1 0 0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 5 and 25.
Its isogeny class 1342.b consists of 3 curves linked by isogenies of degrees dividing 25.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{5}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.3.5368.1 $$\Z/10\Z$$ Not in database
5 5.5.202716958081.1 $$\Z/25\Z$$ Not in database
6 6.6.154681196032.1 $$\Z/2\Z \times \Z/10\Z$$ 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.