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This is the Frey curve for the triple $1 + 80 = 81$ (or in factored form, $1 + 2^4 \cdot 5 = 3^4$).

Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 1, -135, -660]); // or
magma: E := EllipticCurve("15a2");
sage: E = EllipticCurve([1, 1, 1, -135, -660]) # or
sage: E = EllipticCurve("15a2")
gp: E = ellinit([1, 1, 1, -135, -660]) \\ or
gp: E = ellinit("15a2")

$$y^2 + x y + y = x^{3} + x^{2} - 135 x - 660$$

Mordell-Weil group structure

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

Torsion generators

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

$$\left(-7, 3\right)$$, $$\left(13, -7\right)$$

Integral points

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

$$\left(-7, 3\right)$$, $$\left(13, -7\right)$$

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

Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E) Conductor: $$15$$ = $$3 \cdot 5$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$164025$$ = $$3^{8} \cdot 5^{2}$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{272223782641}{164025}$$ = $$3^{-8} \cdot 5^{-2} \cdot 6481^{3}$$ 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 Real period: $$1.40060304233$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]] Tamagawa product: $$4$$  = $$2\cdot2$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E) Torsion order: $$4$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

Modular invariants

Modular form15.2.a.a

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

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

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

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 2 $$\Gamma_0(N)$$-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/factorial(ar)

$$L(E,1)$$ ≈ $$0.350150760583$$

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$$(j)_{-}$$
$$3$$ $$2$$ $$I_{8}$$ Non-split multiplicative 1 1 8 8
$$5$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2

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 X185h.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 0 & 7 \end{array}\right),\left(\begin{array}{rr} 5 & 2 \\ 0 & 1 \end{array}\right)$ and has index 96.

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
$$2$$ Cs

$p$-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 $$0$$.

Iwasawa invariants

$p$ Reduction type $\lambda$-invariant(s) 2 3 5 ordinary nonsplit split 0 0 1 2 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4 and 8.
Its isogeny class 15a consists of 8 curves linked by isogenies of degrees dividing 16.

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

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{-5})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{-1})$$ $$\Z/2\Z \times \Z/4\Z$$ 2.0.4.1-225.2-a8
$$\Q(\sqrt{5})$$ $$\Z/2\Z \times \Z/4\Z$$ 2.2.5.1-45.1-a7
4 $$\Q(\zeta_{8})$$ $$\Z/2\Z \times \Z/8\Z$$ Not in database
$$\Q(i, \sqrt{10})$$ $$\Z/2\Z \times \Z/8\Z$$ Not in database
$$\Q(i, \sqrt{5})$$ $$\Z/4\Z \times \Z/4\Z$$ Not in database
4.2.2000.1 $$\Z/2\Z \times \Z/8\Z$$ 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.