Properties

 Label 690.g2 Conductor $690$ Discriminant $-331200$ j-invariant $$\frac{6967871}{331200}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, 4, 29])

gp: E = ellinit([1, 1, 1, 4, 29])

magma: E := EllipticCurve([1, 1, 1, 4, 29]);

$$y^2+xy+y=x^3+x^2+4x+29$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(-1, 5\right)$$ $\hat{h}(P)$ ≈ $0.17817856783292533319191289132$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-3, 1\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-3, 1\right)$$, $$\left(-1, 5\right)$$, $$\left(-1, -5\right)$$, $$\left(1, 5\right)$$, $$\left(1, -7\right)$$, $$\left(3, 7\right)$$, $$\left(3, -11\right)$$, $$\left(9, 25\right)$$, $$\left(9, -35\right)$$, $$\left(29, 145\right)$$, $$\left(29, -175\right)$$, $$\left(51, 343\right)$$, $$\left(51, -395\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$690$$ = $2 \cdot 3 \cdot 5 \cdot 23$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-331200$ = $-1 \cdot 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 23$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{6967871}{331200}$$ = $2^{-6} \cdot 3^{-2} \cdot 5^{-2} \cdot 23^{-1} \cdot 191^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.26104138516639260158036720340\dots$ Stable Faltings height: $-0.26104138516639260158036720340\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.17817856783292533319191289132\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $2.3108328098155996871753465250\dots$ 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: $24$  = $( 2 \cdot 3 )\cdot2\cdot2\cdot1$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $2$ 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: $L'(E,1)$ ≈ $2.4704452833256696474380849929013852133$

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 96 $\Gamma_0(N)$-optimal: yes Manin constant: 1

Local data

This elliptic curve is 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$(j)_{-}$
$2$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$3$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$5$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$23$ $1$ $I_{1}$ 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 $\GL(2,\Z_\ell)$ for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 2.3.0.1

$p$-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 nonsplit nonsplit ordinary ordinary ordinary ordinary ss split ordinary ss ordinary ordinary ordinary ss 2 1 1 1 1 1 1 1,1 2 1 1,1 1 1 1 1,1 0 0 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 $d=$ 2.
Its isogeny class 690.g consists of 2 curves linked by isogenies of degree 2.

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

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-23})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $4$ 4.2.13248.1 $$\Z/4\Z$$ Not in database $8$ 8.0.92844527616.6 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.2.30983121016875.6 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ 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.