# Properties

 Label 50.a2 Conductor 50 Discriminant -12500000 j-invariant $$-\frac{121945}{32}$$ CM no Rank 0 Torsion Structure $$\Z/{3}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 1, -76, 298]); // or
magma: E := EllipticCurve("50a3");
sage: E = EllipticCurve([1, 0, 1, -76, 298]) # or
sage: E = EllipticCurve("50a3")
gp: E = ellinit([1, 0, 1, -76, 298]) \\ or
gp: E = ellinit("50a3")

$$y^2 + x y + y = x^{3} - 76 x + 298$$

## Mordell-Weil group structure

$$\Z/{3}\Z$$

## Torsion generators

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

$$\left(2, 11\right)$$

## Integral points

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

$$\left(2, 11\right)$$

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$50$$ = $$2 \cdot 5^{2}$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$-12500000$$ = $$-1 \cdot 2^{5} \cdot 5^{8}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$-\frac{121945}{32}$$ = $$-1 \cdot 2^{-5} \cdot 5 \cdot 29^{3}$$ 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: $$2.13949494428$$ 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: $$3$$  = $$1\cdot3$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] Torsion order: $$3$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form50.2.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)

$$q - q^{2} + q^{3} + q^{4} - q^{6} + 2q^{7} - q^{8} - 2q^{9} - 3q^{11} + q^{12} - 4q^{13} - 2q^{14} + q^{16} - 3q^{17} + 2q^{18} + 5q^{19} + O(q^{20})$$

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 10 $$\Gamma_0(N)$$-optimal: no Manin constant: not computed

#### 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])

$$L(E,1)$$ ≈ $$0.713164981428$$

## 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$$(j)_{-}$$
$$2$$ $$1$$ $$I_{5}$$ Non-split multiplicative 1 1 5 5
$$5$$ $$3$$ $$IV^{*}$$ Additive -1 2 8 0

## Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X4.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 7 & 7 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 1 & 1 \end{array}\right)$ and has index 2.

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
$$3$$ B.1.1
$$5$$ B.1.4

## $p$-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) 2 3 5 nonsplit ordinary add 2 2 - 0 0 -

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

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, 5 and 15.
Its isogeny class 50.a consists of 4 curves linked by isogenies of degrees dividing 15.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{5})$$ $$\Z/15\Z$$ 2.2.5.1-100.1-b2
3 3.1.200.1 $$\Z/6\Z$$ Not in database
6 6.0.270000.1 $$\Z/3\Z \times \Z/3\Z$$ Not in database
6.0.320000.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database
6.2.200000.1 $$\Z/30\Z$$ 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.