# Properties

 Label 1470.l1 Conductor $1470$ Discriminant $-553190400000$ j-invariant $$-\frac{1231272543361}{230400000}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, -2990, 71147]) # or

sage: E = EllipticCurve("1470l1")

gp: E = ellinit([1, 1, 1, -2990, 71147]) \\ or

gp: E = ellinit("1470l1")

magma: E := EllipticCurve([1, 1, 1, -2990, 71147]); // or

magma: E := EllipticCurve("1470l1");

$$y^2 + x y + y = x^{3} + x^{2} - 2990 x + 71147$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(27, -119\right)$$ $$\hat{h}(P)$$ ≈ $0.009964107999912606$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-63, 151\right)$$, $$\left(-63, -89\right)$$, $$\left(-43, 371\right)$$, $$\left(-43, -329\right)$$, $$\left(-15, 343\right)$$, $$\left(-15, -329\right)$$, $$\left(7, 221\right)$$, $$\left(7, -229\right)$$, $$\left(17, 151\right)$$, $$\left(17, -169\right)$$, $$\left(27, 91\right)$$, $$\left(27, -119\right)$$, $$\left(41, 119\right)$$, $$\left(41, -161\right)$$, $$\left(45, 151\right)$$, $$\left(45, -197\right)$$, $$\left(57, 271\right)$$, $$\left(57, -329\right)$$, $$\left(97, 791\right)$$, $$\left(97, -889\right)$$, $$\left(167, 1981\right)$$, $$\left(167, -2149\right)$$, $$\left(237, 3451\right)$$, $$\left(237, -3689\right)$$, $$\left(657, 16471\right)$$, $$\left(657, -17129\right)$$, $$\left(1457, 54871\right)$$, $$\left(1457, -56329\right)$$, $$\left(6075, 470491\right)$$, $$\left(6075, -476567\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$1470$$ = $$2 \cdot 3 \cdot 5 \cdot 7^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-553190400000$$ = $$-1 \cdot 2^{13} \cdot 3^{2} \cdot 5^{5} \cdot 7^{4}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{1231272543361}{230400000}$$ = $$-1 \cdot 2^{-13} \cdot 3^{-2} \cdot 5^{-5} \cdot 7^{2} \cdot 29^{3} \cdot 101^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.00996410799991$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.885884712468$$ 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: $$390$$  = $$13\cdot2\cdot5\cdot3$$ 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)

## 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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 3120 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

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);

$$L'(E,1)$$ ≈ $$3.4425498707$$

## Local data

This elliptic curve is not semistable.

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$$(j)_{-}$$
$$2$$ $$13$$ $$I_{13}$$ Split multiplicative -1 1 13 13
$$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$5$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$7$$ $$3$$ $$IV$$ Additive 1 2 4 0

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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 mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ .

## $p$-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 split nonsplit split add ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary 2 1 4 - 1 1 1 1 1 1,1 1 1 1 1 1 0 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 no rational isogenies. Its isogeny class 1470.l consists of this curve only.

## 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
$3$ 3.1.1960.1 $$\Z/2\Z$$ Not in database
$6$ 6.0.153664000.2 $$\Z/2\Z \times \Z/2\Z$$ 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.