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This isogeny class and its quadratic twist by $\Q(\sqrt{-3})$ are the ones of minimal conductor with a $13$-isogeny.

Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 1, -2, -1])

gp: E = ellinit([0, -1, 1, -2, -1])

magma: E := EllipticCurve([0, -1, 1, -2, -1]);

$$y^2+y=x^3-x^2-2x-1$$ trivial

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$147$$ = $3 \cdot 7^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-147$ = $-1 \cdot 3 \cdot 7^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{28672}{3}$$ = $-1 \cdot 2^{12} \cdot 3^{-1} \cdot 7$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.84508751641905230202498515193\dots$ Stable Faltings height: $-1.1694058745949378528758772758\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $1.9200537453490419433285021703\dots$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $1$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $1$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (exact) sage: r = E.rank(); sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: ar = ellanalyticrank(E); gp: ar/factorial(ar)  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); Special value: $L(E,1)$ ≈ $1.9200537453490419433285021702840752175$

Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q + 2 q^{2} - q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + q^{9} + 4 q^{10} - 2 q^{11} - 2 q^{12} - q^{13} - 2 q^{15} - 4 q^{16} + 2 q^{18} - q^{19} + O(q^{20})$$ sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 6 $\Gamma_0(N)$-optimal: yes Manin constant: 1

Local data

This elliptic curve is not semistable. There are 2 primes of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$3$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$7$ $1$ $II$ Additive -1 2 2 0

Galois representations

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 $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$13$ 13B.4.1 13.28.0.1

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 ss nonsplit ordinary add ordinary ordinary ss ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary 0,1 0 0 - 0 0 0,0 0 0,0 0 0 0 0 0 0 0,0 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 non-trivial cyclic isogenies of degree $d$ for $d=$ 13.
Its isogeny class 147c consists of 2 curves linked by isogenies of degree 13.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.588.1 $$\Z/2\Z$$ Not in database $6$ 6.0.1037232.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ $$\Q(\zeta_{7})$$ $$\Z/13\Z$$ Not in database $8$ 8.2.20841167403.1 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $18$ 18.0.14176142707705638912.1 $$\Z/26\Z$$ Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.