Properties

 Label 5610q1 Conductor $5610$ Discriminant $-2524500$ j-invariant $$-\frac{119168121961}{2524500}$$ CM no Rank $1$ Torsion structure $$\Z/{3}\Z$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -103, 398]) # or

sage: E = EllipticCurve("5610q1")

gp: E = ellinit([1, 0, 1, -103, 398]) \\ or

gp: E = ellinit("5610q1")

magma: E := EllipticCurve([1, 0, 1, -103, 398]); // or

magma: E := EllipticCurve("5610q1");

$$y^2 + x y + y = x^{3} - 103 x + 398$$

Mordell-Weil group structure

$$\Z\times \Z/{3}\Z$$

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(-11, 20\right)$$ $$\hat{h}(P)$$ ≈ $0.6859974733109373$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(4, 5\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-11, 20\right)$$, $$\left(-11, -10\right)$$, $$\left(4, 5\right)$$, $$\left(4, -10\right)$$, $$\left(7, 2\right)$$, $$\left(7, -10\right)$$, $$\left(9, 10\right)$$, $$\left(9, -20\right)$$, $$\left(94, 860\right)$$, $$\left(94, -955\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$5610$$ = $$2 \cdot 3 \cdot 5 \cdot 11 \cdot 17$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-2524500$$ = $$-1 \cdot 2^{2} \cdot 3^{3} \cdot 5^{3} \cdot 11 \cdot 17$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{119168121961}{2524500}$$ = $$-1 \cdot 2^{-2} \cdot 3^{-3} \cdot 5^{-3} \cdot 7^{3} \cdot 11^{-1} \cdot 17^{-1} \cdot 19^{3} \cdot 37^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$

BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.685997473311$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$2.57040779609$$ 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: $$18$$  = $$2\cdot3\cdot3\cdot1\cdot1$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$3$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

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);

$$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{7} - q^{8} + q^{9} - q^{10} - q^{11} + q^{12} - q^{13} + q^{14} + q^{15} + q^{16} - q^{17} - q^{18} - 7q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1440 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

Special L-value

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);

$$L'(E,1)$$ ≈ $$3.526586507$$

Local data

This elliptic curve is semistable.

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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$3$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$5$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$11$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$17$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1

Galois representations

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

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 mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$3$$ 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 nonsplit split split ordinary nonsplit ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary 1 2 2 1 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

Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 5610q consists of 2 curves linked by isogenies of degree 3.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.1.11220.1 $$\Z/6\Z$$ Not in database
$6$ 6.0.528262975152.2 $$\Z/3\Z \times \Z/3\Z$$ Not in database
$6$ 6.0.1412467848000.1 $$\Z/2\Z \times \Z/6\Z$$ 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.