# Properties

 Label 17.a4 Conductor $17$ Discriminant $17$ j-invariant $$\frac{35937}{17}$$ CM no Rank $0$ Torsion structure $$\Z/{4}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

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

gp: E = ellinit([1, -1, 1, -1, 0])

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

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

## Mordell-Weil group structure

$\Z/{4}\Z$

## 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);

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

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$17$$ = $17$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $17$ = $17$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{35937}{17}$$ = $3^{3} \cdot 11^{3} \cdot 17^{-1}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-1.0697833015012887903746651257\dots$ Stable Faltings height: $-1.0697833015012887903746651257\dots$

## 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[1]  magma: RealPeriod(E); Real period: $6.1883190142044806928383160199\dots$ 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: $1$ sage: E.torsion_order()  gp: elltors(E)[1]  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[2]/factorial(ar[1])  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); Special value: $L(E,1)$ ≈ $0.38676993838778004330239475124309554740$

## 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);

$$q - q^{2} - q^{4} - 2q^{5} + 4q^{7} + 3q^{8} - 3q^{9} + 2q^{10} - 2q^{13} - 4q^{14} - q^{16} + q^{17} + 3q^{18} - 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 4 $\Gamma_0(N)$-optimal: no Manin constant: 4

## Local data

This elliptic curve is semistable. There is only one prime 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$(j)_{-}$
$17$ $1$ $I_{1}$ 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 $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 32.96.0.10

## $p$-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 2 17 ordinary split 0 1 0 0

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

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 17.a consists of 3 curves linked by isogenies of degrees dividing 4.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{17})$$ $$\Z/2\Z \times \Z/4\Z$$ 2.2.17.1-17.1-a4 $4$ 4.0.272.1 $$\Z/8\Z$$ Not in database $4$ 4.4.4913.1 $$\Z/2\Z \times \Z/8\Z$$ 4.4.4913.1-17.1-a8 $8$ 8.0.6179217664.3 $$\Z/4\Z \times \Z/4\Z$$ Not in database $8$ 8.0.21381376.2 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ 8.2.182660427.1 $$\Z/12\Z$$ Not in database $16$ 16.0.38182730939089616896.1 $$\Z/4\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ 16.8.43104723599206637824.1 $$\Z/2\Z \times \Z/16\Z$$ 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.