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## Minimal Weierstrass equation

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

gp: E = ellinit([0, 1, 0, -1, 0])

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

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

$$\Z/{6}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-1, 1\right)$$ ## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

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

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$20$$ = $$2^{2} \cdot 5$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$80$$ = $$2^{4} \cdot 5$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{16384}{5}$$ = $$2^{14} \cdot 5^{-1}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$-0.92995009551551997117402608057\dots$$ Stable Faltings height: $$-1.1609991557021684076464367877\dots$$

## 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: $$5.6487502839182275989675790981\dots$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$3$$  = $$3\cdot1$$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $$6$$ 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);

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

#### 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);

$$L(E,1)$$ ≈ $$0.47072919032651896658063159150721550180$$

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

## Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X9c.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 3 \end{array}\right)$ and has index 12.

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
$$2$$ B
$$3$$ 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) 2 3 5 add ordinary nonsplit - 2 0 - 0 0

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

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 $$d=$$ 2, 3 and 6.
Its isogeny class 20.a consists of 4 curves linked by isogenies of degrees dividing 6.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{5})$$ $$\Z/2\Z \times \Z/6\Z$$ 2.2.5.1-80.1-a4 $4$ 4.0.320.1 $$\Z/12\Z$$ Not in database $6$ 6.0.270000.1 $$\Z/3\Z \times \Z/6\Z$$ Not in database $8$ 8.0.2560000.1 $$\Z/2\Z \times \Z/12\Z$$ Not in database $8$ 8.4.64000000.1 $$\Z/2\Z \times \Z/12\Z$$ Not in database $9$ 9.3.787320000.1 $$\Z/18\Z$$ Not in database $12$ 12.0.1822500000000.1 $$\Z/6\Z \times \Z/6\Z$$ Not in database $16$ 16.0.4096000000000000.1 $$\Z/4\Z \times \Z/12\Z$$ Not in database $16$ 16.0.1717986918400000000.1 $$\Z/24\Z$$ Not in database $18$ 18.6.1937102445000000000000.1 $$\Z/2\Z \times \Z/18\Z$$ Not in database

We only show fields where the torsion growth is primitive.