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## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -2255, 19622])

gp: E = ellinit([1, -1, 1, -2255, 19622])

magma: E := EllipticCurve([1, -1, 1, -2255, 19622]);

$$y^2+xy+y=x^3-x^2-2255x+19622$$ ## Mordell-Weil group structure

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

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-51, 25\right)$$, $$\left(9, -5\right)$$ ## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-51, 25\right)$$, $$\left(9, -5\right)$$ ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$225$$ = $3^{2} \cdot 5^{2}$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $576650390625$ = $3^{10} \cdot 5^{10}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{111284641}{50625}$$ = $3^{-4} \cdot 5^{-4} \cdot 13^{3} \cdot 37^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.95174752846826124555440578791\dots$ Stable Faltings height: $-0.40227757208284378744359649716\dots$

## 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: $0.82429593904393569890541103843\dots$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $16$  = $2^{2}\cdot2^{2}$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $4$ 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: $L(E,1)$ ≈ $0.82429593904393569890541103843228277079$

## 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);

$$q - q^{2} - q^{4} + 3 q^{8} + 4 q^{11} + 2 q^{13} - q^{16} + 2 q^{17} + 4 q^{19} + O(q^{20})$$ sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 192 $\Gamma_0(N)$-optimal: no 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$(j)_{-}$
$3$ $4$ $I_{4}^{*}$ Additive -1 2 10 4
$5$ $4$ $I_{4}^{*}$ Additive 1 2 10 4

## 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
$2$ 2Cs 8.96.0.61

## $p$-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 ordinary add add ? - - ? - -

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

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 and 4.
Its isogeny class 225c consists of 3 curves linked by isogenies of degrees dividing 16.

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

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-15})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{15})$$ $$\Z/2\Z \times \Z/4\Z$$ 2.2.60.1-15.1-a5 $4$ $$\Q(i, \sqrt{15})$$ $$\Z/4\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{6}, \sqrt{10})$$ $$\Z/2\Z \times \Z/8\Z$$ Not in database $4$ $$\Q(\sqrt{-3}, \sqrt{5})$$ $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.0.12960000.1 $$\Z/4\Z \times \Z/8\Z$$ Not in database $8$ 8.0.3317760000.5 $$\Z/4\Z \times \Z/8\Z$$ Not in database $8$ 8.2.2767921875.2 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ 16.0.11007531417600000000.1 $$\Z/8\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/16\Z$$ Not in database $16$ 16.0.104976000000000000.4 $$\Z/2\Z \times \Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ 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.