# Properties

 Label 665.a2 Conductor 665 Discriminant -18960396875 j-invariant $$-\frac{1029077364736}{18960396875}$$ CM no Rank 1 Torsion Structure $$\Z/{5}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, -1, 1, -210, 6798]); // or
magma: E := EllipticCurve("665d1");
sage: E = EllipticCurve([0, -1, 1, -210, 6798]) # or
sage: E = EllipticCurve("665d1")
gp: E = ellinit([0, -1, 1, -210, 6798]) \\ or
gp: E = ellinit("665d1")

$$y^2 + y = x^{3} - x^{2} - 210 x + 6798$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

 $$P$$ = $$\left(-18, 66\right)$$ $$\hat{h}(P)$$ ≈ 0.523665501103

## Torsion generators

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

$$\left(-11, 87\right)$$

## Integral points

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

$$\left(-18, 66\right)$$, $$\left(-11, 87\right)$$, $$\left(4, 77\right)$$, $$\left(24, 122\right)$$, $$\left(39, 237\right)$$, $$\left(73, 612\right)$$, $$\left(129, 1452\right)$$, $$\left(514, 11637\right)$$

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$665$$ = $$5 \cdot 7 \cdot 19$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$-18960396875$$ = $$-1 \cdot 5^{5} \cdot 7^{5} \cdot 19^{2}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$-\frac{1029077364736}{18960396875}$$ = $$-1 \cdot 2^{12} \cdot 5^{-5} \cdot 7^{-5} \cdot 19^{-2} \cdot 631^{3}$$ 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: $$0.523665501103$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$1.0294222183$$ 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: $$50$$  = $$5\cdot5\cdot2$$ 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 form665.2.a.a

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)

$$q - 2q^{2} - q^{3} + 2q^{4} + q^{5} + 2q^{6} + q^{7} - 2q^{9} - 2q^{10} - 3q^{11} - 2q^{12} - q^{13} - 2q^{14} - q^{15} - 4q^{16} + 3q^{17} + 4q^{18} - q^{19} + O(q^{20})$$

#### Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
600 . This curve is $$\Gamma_0(N)$$-optimal.

#### 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])

$$L'(E,1)$$ ≈ $$1.07814580359$$

## 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$$(j)_{-}$$
$$5$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$7$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$19$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2

## 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]

Note: $$p$$-adic regulator data only exists for primes $$p\ge5$$ 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 ss ordinary split split ordinary ordinary ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary 1,6 5 6 2 1 1 1 1 1 1 1 1 1 1 1 0,0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 5.
Its isogeny class 665.a 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 3.1.140.1 $$\Z/10\Z$$ Not in database
6 6.0.686000.1 $$\Z/2\Z \times \Z/10\Z$$ 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.