Properties

 Label 429429.y4 Conductor 429429 Discriminant 13826945297998833 j-invariant $$\frac{4354703137}{1617}$$ CM no Rank 1 Torsion Structure $$\Z/{2}\Z$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 1, -695692, -223561876]); // or

magma: E := EllipticCurve("429429y1");

sage: E = EllipticCurve([1, 1, 1, -695692, -223561876]) # or

sage: E = EllipticCurve("429429y1")

gp: E = ellinit([1, 1, 1, -695692, -223561876]) \\ or

gp: E = ellinit("429429y1")

$$y^2 + x y + y = x^{3} + x^{2} - 695692 x - 223561876$$

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(\frac{1296511}{25}, \frac{1472849276}{125}\right)$$ $$\hat{h}(P)$$ ≈ 8.14863791757

Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(-489, 244\right)$$

Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-489, 244\right)$$

Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$429429$$ = $$3 \cdot 7 \cdot 11^{2} \cdot 13^{2}$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$13826945297998833$$ = $$3 \cdot 7^{2} \cdot 11^{7} \cdot 13^{6}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{4354703137}{1617}$$ = $$3^{-1} \cdot 7^{-2} \cdot 11^{-1} \cdot 23^{3} \cdot 71^{3}$$ 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: $$8.148637917566$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.165312227255272$$ 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: $$16$$  = $$1\cdot2\cdot2^{2}\cdot2$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$2$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

Modular invariants

Modular form 429429.2.a.y

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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 4608000 $$\Gamma_0(N)$$-optimal: unknown* (one of 4 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 400000. 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[2]/factorial(ar[1])

$$L'(E,1)$$ ≈ $$5.388277932998397$$

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$$(j)_{-}$$
$$3$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$7$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$11$$ $$4$$ $$I_1^{*}$$ Additive -1 2 7 1
$$13$$ $$2$$ $$I_0^{*}$$ Additive 1 2 6 0

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 X36.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$ and has index 12.

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, 4 and 8.
Its isogeny class 429429.y consists of 6 curves linked by isogenies of degrees dividing 8.