# Properties

 Label 8550v1 Conductor $8550$ Discriminant $-68080435200$ j-invariant $$\frac{272199695}{3735552}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, 355, 12197])

gp: E = ellinit([1, -1, 1, 355, 12197])

magma: E := EllipticCurve([1, -1, 1, 355, 12197]);

$$y^2+xy+y=x^3-x^2+355x+12197$$

## Mordell-Weil group structure

$\Z$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(15, 136\right)$$ $\hat{h}(P)$ ≈ $0.24905264924815378357041467727$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-17, 40\right)$$, $$\left(-17, -24\right)$$, $$\left(15, 136\right)$$, $$\left(15, -152\right)$$, $$\left(33, 226\right)$$, $$\left(33, -260\right)$$, $$\left(415, 8248\right)$$, $$\left(415, -8664\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$8550$$ = $2 \cdot 3^{2} \cdot 5^{2} \cdot 19$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-68080435200$ = $-1 \cdot 2^{16} \cdot 3^{7} \cdot 5^{2} \cdot 19$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{272199695}{3735552}$$ = $2^{-16} \cdot 3^{-1} \cdot 5 \cdot 19^{-1} \cdot 379^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $0.75940167429530358746552500909\dots$ Stable Faltings height: $-0.058144122111101320665557498242\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.24905264924815378357041467727\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.81378976942485158827633474387\dots$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $32$  = $2^{4}\cdot2\cdot1\cdot1$ sage: E.torsion_order()  gp: elltors(E)[1]  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[2]/factorial(ar[1])  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); Special value: $L'(E,1)$ ≈ $6.4856479362017121537840254883814628277$

## Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

magma: ModularForm(E);

$$q + q^{2} + q^{4} + q^{8} - q^{11} + 4q^{13} + q^{16} - 4q^{17} - q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 6144 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 4 primes of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)[5]

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

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $16$ $I_{16}$ Split multiplicative -1 1 16 16
$3$ $2$ $I_1^{*}$ Additive -1 2 7 1
$5$ $1$ $II$ Additive 1 2 2 0
$19$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

## 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 $\ell$.

## $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

## 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 split add add ss ordinary ordinary ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ss 11 - - 1,3 1 1 3 1 1 3 1 1 1 1 1,1 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 no rational isogenies. Its isogeny class 8550v consists of this curve only.

## 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 $K$ $3$ 3.1.5700.1 $$\Z/2\Z$$ Not in database $6$ 6.0.7407720000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ 8.2.5771493546750000.11 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ 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.