# Properties

 Label 606.f1 Conductor 606 Discriminant -785376 j-invariant $$-\frac{80677568161}{785376}$$ CM no Rank 0 Torsion Structure $$\Z/{5}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -90, 324]); // or
magma: E := EllipticCurve("606f1");
sage: E = EllipticCurve([1, 0, 0, -90, 324]) # or
sage: E = EllipticCurve("606f1")
gp: E = ellinit([1, 0, 0, -90, 324]) \\ or
gp: E = ellinit("606f1")

$$y^2 + x y = x^{3} - 90 x + 324$$

## Mordell-Weil group structure

$$\Z/{5}\Z$$

## Torsion generators

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

$$\left(0, 18\right)$$

## Integral points

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

$$\left(0, 18\right)$$, $$\left(6, 0\right)$$

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$606$$ = $$2 \cdot 3 \cdot 101$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$-785376$$ = $$-1 \cdot 2^{5} \cdot 3^{5} \cdot 101$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$-\frac{80677568161}{785376}$$ = $$-1 \cdot 2^{-5} \cdot 3^{-5} \cdot 29^{3} \cdot 101^{-1} \cdot 149^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E); sage: E.rank() Rank: $$0$$ magma: Regulator(E); sage: E.regulator() Regulator: $$1$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$2.84676247014$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$25$$  = $$5\cdot5\cdot1$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] Torsion order: $$5$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form606.2.a.f

magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

$$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - 2q^{7} + q^{8} + q^{9} + q^{10} + 2q^{11} + q^{12} + 4q^{13} - 2q^{14} + q^{15} + q^{16} - 2q^{17} + q^{18} + O(q^{20})$$

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 100 $$\Gamma_0(N)$$-optimal: yes 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[2]/factorial(ar[1])

$$L(E,1)$$ ≈ $$2.84676247014$$

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(E)[5]
prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$3$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$101$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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
$$5$$ B.1.1

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) 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) $\mu$-invariant(s) 2 3 5 101 split split ordinary split 6 3 2 1 0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 5.
Its isogeny class 606.f consists of 2 curves linked by isogenies of degree 5.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{5}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.2424.1 $$\Z/10\Z$$ Not in database
6 6.0.14242881024.1 $$\Z/2\Z \times \Z/10\Z$$ 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.