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## Minimal Weierstrass equation

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

gp: E = ellinit([0, 1, 1, -1, 0])

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

$$y^2+y=x^3+x^2-x$$ ## Mordell-Weil group structure

$$\Z/{3}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(0, 0\right)$$ ## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(0, 0\right)$$, $$\left(0, -1\right)$$ ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$35$$ = $$5 \cdot 7$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-35$$ = $$-1 \cdot 5 \cdot 7$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{262144}{35}$$ = $$-1 \cdot 2^{18} \cdot 5^{-1} \cdot 7^{-1}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$-0.97115016503710495708757733528\dots$$ Stable Faltings height: $$-0.97115016503710495708757733528\dots$$

## 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: $$6.3262011522141496364219629784\dots$$ 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: $$3$$ 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)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q + q^{3} - 2q^{4} - q^{5} + q^{7} - 2q^{9} - 3q^{11} - 2q^{12} + 5q^{13} - q^{15} + 4q^{16} + 3q^{17} + 2q^{19} + O(q^{20})$$ sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 6 $$\Gamma_0(N)$$-optimal: no Manin constant: 3

#### Special L-value

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);

$$L(E,1)$$ ≈ $$0.70291123913490551515799588648696046342$$

## Local data

This elliptic curve is 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$$(j)_{-}$$
$$5$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$7$$ $$1$$ $$I_{1}$$ 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
$$3$$ B.1.1

## $p$-adic data

### $p$-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 ss ordinary nonsplit split 0,5 0 0 1 0,0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3 and 9.
Its isogeny class 35.a consists of 3 curves linked by isogenies of degrees dividing 9.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $3$ 3.1.140.1 $$\Z/6\Z$$ Not in database $3$ $$\Q(\zeta_{7})^+$$ $$\Z/9\Z$$ 3.3.49.1-875.1-a2 $6$ 6.0.686000.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database $6$ 6.0.40516875.1 $$\Z/3\Z \times \Z/3\Z$$ Not in database $6$ 6.0.826875.2 $$\Z/9\Z$$ Not in database $9$ 9.3.6588344000.1 $$\Z/18\Z$$ Not in database $12$ Deg 12 $$\Z/12\Z$$ Not in database $18$ 18.0.66513197260078857421875.1 $$\Z/3\Z \times \Z/9\Z$$ Not in database $18$ 18.0.333736618572171675000000000000.1 $$\Z/3\Z \times \Z/6\Z$$ Not in database $18$ 18.0.138999008151675000000000000.1 $$\Z/18\Z$$ Not in database $18$ 18.0.37980492079544000000000.1 $$\Z/2\Z \times \Z/18\Z$$ 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.