Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 1, -1632, 104769]); // or
magma: E := EllipticCurve("361722u1");
sage: E = EllipticCurve([1, 1, 1, -1632, 104769]) # or
sage: E = EllipticCurve("361722u1")
gp: E = ellinit([1, 1, 1, -1632, 104769]) \\ or
gp: E = ellinit("361722u1")

$$y^2 + x y + y = x^{3} + x^{2} - 1632 x + 104769$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

 $$P$$ = $$\left(-21, 371\right)$$ $$\hat{h}(P)$$ ≈ 0.862854773843

Torsion generators

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

$$\left(-59, 29\right)$$

Integral points

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

$$\left(-59, 29\right)$$, $$\left(-21, 371\right)$$, $$\left(41, 309\right)$$, $$\left(169, 2081\right)$$

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

Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E) Conductor: $$361722$$ = $$2 \cdot 3 \cdot 19^{2} \cdot 167$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$-4525437385152$$ = $$-1 \cdot 2^{6} \cdot 3^{2} \cdot 19^{6} \cdot 167$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$-\frac{10218313}{96192}$$ = $$-1 \cdot 2^{-6} \cdot 3^{-2} \cdot 7^{3} \cdot 31^{3} \cdot 167^{-1}$$ 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: $$0.862854773843$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega Real period: $$0.661321906494$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$48$$  = $$( 2 \cdot 3 )\cdot2\cdot2^{2}\cdot1$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E) Torsion order: $$2$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

Modular invariants

Modular form 361722.2.a.u

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

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

For more coefficients, see the Downloads section to the right.

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 995328 $$\Gamma_0(N)$$-optimal: unknown* (one of 2 curves in this isogeny class which might be optimal) Manin constant: 1 (conditional*)
* The optimal curve in each isogeny class has not been determined in all cases for conductors over 300000. The Manin constant is correct provided that this curve is 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/factorial(ar)

$$L'(E,1)$$ ≈ $$6.84749716878$$

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(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
$$19$$ $$4$$ $$I_0^{*}$$ Additive -1 2 6 0
$$167$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X6.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right)$ and has index 3.

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
$$2$$ B

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

$$p$$-adic regulators are not yet computed for curves that are not $$\Gamma_0$$-optimal.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 361722u consists of 2 curves linked by isogenies of degree 2.

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

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{-167})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
4 $$x^{4}$$ $$\mathstrut -\mathstrut 160 x^{2}$$ $$\mathstrut -\mathstrut 2052 x$$ $$\mathstrut -\mathstrut 16685$$ $$\Z/4\Z$$ 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.