# Properties

 Label 71058.m2 Conductor 71058 Discriminant 2048285853033578496 j-invariant $$\frac{14828809162780998661814929}{2048285853033578496}$$ CM no Rank 1 Torsion Structure $$\Z/{7}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -5118321, 4456008297]) # or

sage: E = EllipticCurve("71058l1")

gp: E = ellinit([1, 0, 0, -5118321, 4456008297]) \\ or

gp: E = ellinit("71058l1")

magma: E := EllipticCurve([1, 0, 0, -5118321, 4456008297]); // or

magma: E := EllipticCurve("71058l1");

$$y^2 + x y = x^{3} - 5118321 x + 4456008297$$

## Mordell-Weil group structure

$$\Z\times \Z/{7}\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-1794, 89583\right)$$ $$\hat{h}(P)$$ ≈ 3.1446330531863507

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(1326, 351\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-1794, 89583\right)$$, $$\left(-1794, -87789\right)$$, $$\left(-1378, 94991\right)$$, $$\left(-1378, -93613\right)$$, $$\left(1242, 3291\right)$$, $$\left(1242, -4533\right)$$, $$\left(1326, 351\right)$$, $$\left(1326, -1677\right)$$, $$\left(1482, 10491\right)$$, $$\left(1482, -11973\right)$$, $$\left(3354, 156507\right)$$, $$\left(3354, -159861\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$71058$$ = $$2 \cdot 3 \cdot 13 \cdot 911$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$2048285853033578496$$ = $$2^{14} \cdot 3^{7} \cdot 13^{7} \cdot 911$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{14828809162780998661814929}{2048285853033578496}$$ = $$2^{-14} \cdot 3^{-7} \cdot 13^{-7} \cdot 157^{3} \cdot 911^{-1} \cdot 1564837^{3}$$ Endomorphism ring: $$\Z$$ (no 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: $$3.14463305319$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.252325060397$$ 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: $$686$$  = $$( 2 \cdot 7 )\cdot7\cdot7\cdot1$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$7$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 71058.2.a.m

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2304960 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$11.108576151$$

## Local data

This elliptic curve is 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$$(j)_{-}$$
$$2$$ $$14$$ $$I_{14}$$ Split multiplicative -1 1 14 14
$$3$$ $$7$$ $$I_{7}$$ Split multiplicative -1 1 7 7
$$13$$ $$7$$ $$I_{7}$$ Split multiplicative -1 1 7 7
$$911$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1

## 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 $$p$$ except those listed.

prime Image of Galois representation
$$7$$ 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]

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 47 911 split split ordinary ordinary ordinary split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary split 2 2 3 3 5 2 1 1 1 1 1 1 1 1 1 2 0 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 $$d=$$ 7.
Its isogeny class 71058.m 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 $E(\Q)_{\rm tors}$ $\cong \Z/{7}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.3.35529.1 $$\Z/14\Z$$ Not in database
6 6.6.44848606340889.1 $$\Z/2\Z \times \Z/14\Z$$ 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.