Properties

 Label 176505s3 Conductor 176505 Discriminant -4619025114469605 j-invariant $$\frac{590589719}{972405}$$ CM no Rank 0 Torsion Structure $$\Z/{2}\Z$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 0, 29383, -2621034]); // or

magma: E := EllipticCurve("176505s3");

sage: E = EllipticCurve([1, 1, 0, 29383, -2621034]) # or

sage: E = EllipticCurve("176505s3")

gp: E = ellinit([1, 1, 0, 29383, -2621034]) \\ or

gp: E = ellinit("176505s3")

$$y^2 + x y = x^{3} + x^{2} + 29383 x - 2621034$$

Mordell-Weil group structure

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

Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(\frac{299}{4}, -\frac{299}{8}\right)$$

Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

None

Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$176505$$ = $$3 \cdot 5 \cdot 7 \cdot 41^{2}$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-4619025114469605$$ = $$-1 \cdot 3^{4} \cdot 5 \cdot 7^{4} \cdot 41^{6}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{590589719}{972405}$$ = $$3^{-4} \cdot 5^{-1} \cdot 7^{-4} \cdot 839^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$0$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$1$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.22892939588$$ 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: $$16$$  = $$2\cdot1\cdot2\cdot2^{2}$$ 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: $$4$$ (exact)

Modular invariants

Modular form 176505.2.a.r

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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 1105920 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

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])

$$L(E,1)$$ ≈ $$3.66287033408$$

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$$(j)_{-}$$
$$3$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$5$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$7$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$41$$ $$4$$ $$I_0^{*}$$ 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 X13.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right)$ and has index 6.

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]

All $$p$$-adic regulators are identically $$1$$ since the rank is $$0$$.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 176505s consists of 4 curves linked by isogenies of degrees dividing 4.

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 $$\Q(\sqrt{-205})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{41})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{-5})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
4 $$\Q(\sqrt{-5}, \sqrt{41})$$ $$\Z/2\Z \times \Z/4\Z$$ 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.