# Properties

 Label 43190s2 Conductor 43190 Discriminant -2382836194386693393110 j-invariant $$-\frac{226953328047600468451761}{2382836194386693393110}$$ CM no Rank 1 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, -1270782, -2412118701]); // or

magma: E := EllipticCurve("43190s2");

sage: E = EllipticCurve([1, -1, 1, -1270782, -2412118701]) # or

sage: E = EllipticCurve("43190s2")

gp: E = ellinit([1, -1, 1, -1270782, -2412118701]) \\ or

gp: E = ellinit("43190s2")

$$y^2 + x y + y = x^{3} - x^{2} - 1270782 x - 2412118701$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(\frac{247621}{144}, \frac{36578569}{1728}\right)$$ $$\hat{h}(P)$$ ≈ 12.8692624379

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

None

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$43190$$ = $$2 \cdot 5 \cdot 7 \cdot 617$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-2382836194386693393110$$ = $$-1 \cdot 2 \cdot 5 \cdot 7 \cdot 617^{7}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{226953328047600468451761}{2382836194386693393110}$$ = $$-1 \cdot 2^{-1} \cdot 3^{3} \cdot 5^{-1} \cdot 7^{-1} \cdot 13^{3} \cdot 43^{3} \cdot 617^{-7} \cdot 36373^{3}$$ 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: $$12.8692624379$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.0616997101948$$ 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: $$7$$  = $$1\cdot1\cdot1\cdot7$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$1$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 43190.2.a.g

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)

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

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 6108144 $$\Gamma_0(N)$$-optimal: no 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])

$$L'(E,1)$$ ≈ $$5.55820833987$$

## 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$$(j)_{-}$$
$$2$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$5$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$7$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$617$$ $$7$$ $$I_{7}$$ Split multiplicative -1 1 7 7

## 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
$$7$$ B.1.3

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) 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 47 617 split ss split split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary split 7 1,3 2 2 1 1 1 1 1 1 1 1 1,1 1 1 2 0 0,0 0 1 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 $$d=$$ 7.
Its isogeny class 43190s consists of 2 curves linked by isogenies of degree 7.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.172760.3 $$\Z/2\Z$$ Not in database
6 $$\Q(\zeta_{7})$$ $$\Z/7\Z$$ Not in database
$$x^{6} - 78 x^{4} + 1521 x^{2} + 172760$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
7 7.1.96889010407000000.8 $$\Z/7\Z$$ 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.