# Properties

 Label 51870p1 Conductor 51870 Discriminant 116188800000 j-invariant $$\frac{12359092816971484921}{116188800000}$$ CM no Rank 1 Torsion Structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 0, -48167, 4048821]); // or
magma: E := EllipticCurve("51870p1");
sage: E = EllipticCurve([1, 1, 0, -48167, 4048821]) # or
sage: E = EllipticCurve("51870p1")
gp: E = ellinit([1, 1, 0, -48167, 4048821]) \\ or
gp: E = ellinit("51870p1")

$$y^2 + x y = x^{3} + x^{2} - 48167 x + 4048821$$

## Mordell-Weil group structure

$$\Z\times \Z/{2}\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

 $$P$$ = $$\left(-98, 2849\right)$$ $$\hat{h}(P)$$ ≈ 0.900373647531

## Torsion generators

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

$$\left(126, -63\right)$$

## Integral points

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

$$\left(-98, 2849\right)$$, $$\left(126, -63\right)$$, $$\left(127, -51\right)$$, $$\left(207, 1629\right)$$

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$51870$$ = $$2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 19$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$116188800000$$ = $$2^{10} \cdot 3 \cdot 5^{5} \cdot 7^{2} \cdot 13 \cdot 19$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{12359092816971484921}{116188800000}$$ = $$2^{-10} \cdot 3^{-1} \cdot 5^{-5} \cdot 7^{-2} \cdot 13^{-1} \cdot 19^{-1} \cdot 103^{3} \cdot 22447^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E); sage: E.rank() Rank: $$1$$ magma: Regulator(E); sage: E.regulator() Regulator: $$0.900373647531$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$0.947961134848$$ 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: $$20$$  = $$2\cdot1\cdot5\cdot2\cdot1\cdot1$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] Torsion order: $$2$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 51870.2.a.p

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)

$$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{7} - q^{8} + q^{9} - q^{10} + 4q^{11} - q^{12} - q^{13} + q^{14} - q^{15} + q^{16} - 6q^{17} - q^{18} + q^{19} + O(q^{20})$$

For more coefficients, see the Downloads section to the right.

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 256000 $$\Gamma_0(N)$$-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])

$$L'(E,1)$$ ≈ $$4.2675961235$$

## 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{10}$$ Non-split multiplicative 1 1 10 10
$$3$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$5$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$7$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$13$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$19$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1

## 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 X6.

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

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
$$2$$ B

## $p$-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 nonsplit nonsplit split nonsplit ordinary nonsplit ordinary split ordinary ordinary ordinary ordinary ordinary ordinary ordinary 4 1 2 3 1 1 3 2 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 51870p consists of 2 curves linked by isogenies of degree 2.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{3705})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
4 $$x^{4}$$ $$\mathstrut -\mathstrut 2 x^{3}$$ $$\mathstrut +\mathstrut 275 x^{2}$$ $$\mathstrut -\mathstrut 274 x$$ $$\mathstrut +\mathstrut 18825$$ $$\Z/4\Z$$ 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.