Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, -313655, 21317222]); // or
magma: E := EllipticCurve("27225bo2");
sage: E = EllipticCurve([1, -1, 1, -313655, 21317222]) # or
sage: E = EllipticCurve("27225bo2")
gp: E = ellinit([1, -1, 1, -313655, 21317222]) \\ or
gp: E = ellinit("27225bo2")

$$y^2 + x y + y = x^{3} - x^{2} - 313655 x + 21317222$$

## Mordell-Weil group structure

$$\Z\times \Z/{2}\Z \times \Z/{2}\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

 $$P$$ = $$\left(-470, 8281\right)$$ $$\hat{h}(P)$$ ≈ 4.2928892923

## Torsion generators

magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp: elltors(E)

$$\left(69, -35\right)$$, $$\left(\frac{2091}{4}, -\frac{2095}{8}\right)$$

## Integral points

magma: IntegralPoints(E);
sage: E.integral_points()

$$\left(-591, 295\right)$$, $$\left(-470, 8281\right)$$, $$\left(69, -35\right)$$, $$\left(5484, 401245\right)$$

Note: only one of each pair $\pm P$ is listed.

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E) Conductor: $$27225$$ = $$3^{2} \cdot 5^{2} \cdot 11^{2}$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$1779985907461265625$$ = $$3^{12} \cdot 5^{6} \cdot 11^{8}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{169112377}{88209}$$ = $$3^{-6} \cdot 7^{3} \cdot 11^{-2} \cdot 79^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E); sage: E.rank() Rank: $$1$$ magma: Regulator(E); sage: E.regulator() Regulator: $$4.2928892923$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega Real period: $$0.232721094979$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$64$$  = $$2^{2}\cdot2^{2}\cdot2^{2}$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E) Torsion order: $$4$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 27225.2.a.r

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

$$q - q^{2} - q^{4} + 4q^{7} + 3q^{8} - 2q^{13} - 4q^{14} - q^{16} + 2q^{17} + O(q^{20})$$

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 368640 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

#### Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
sage: r = E.rank();
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp: ar = ellanalyticrank(E);
gp: ar/factorial(ar)

$$L'(E,1)$$ ≈ $$3.99618358691$$

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(E)
prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$3$$ $$4$$ $$I_6^{*}$$ Additive -1 2 12 6
$$5$$ $$4$$ $$I_0^{*}$$ Additive 1 2 6 0
$$11$$ $$4$$ $$I_2^{*}$$ Additive -1 2 8 2

## 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 X8.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by  and has index 6.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
sage: rho = E.galois_representation();
sage: [rho.image_type(p) for p in rho.non_surjective()]

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

prime Image of Galois representation
$$2$$ Cs

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

$$p$$-adic regulators are not yet computed for curves that are not $$\Gamma_0$$-optimal.

## 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 ordinary add add ordinary add ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ss ordinary ? - - 1 - 1 1 1,1 1 1 1 1 1 1,3 1 ? - - 0 - 0 0 0,0 0 0 0 0 0 0,0 0

An entry ? indicates that the invariants have not yet been computed.

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.
Its isogeny class 27225.r consists of 4 curves linked by isogenies of degrees dividing 4.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
4 $$\Q(\sqrt{3}, \sqrt{-55})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{-11}, \sqrt{-15})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{15}, \sqrt{33})$$ $$\Z/2\Z \times \Z/4\Z$$ 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.