Properties

 Label 84150.bd1 Conductor 84150 Discriminant -2310075281250000 j-invariant $$-\frac{4368317413923}{5475734000}$$ CM no Rank 2 Torsion Structure $$\mathrm{Trivial}$$

Related objects

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 0, -25542, 2802116]); // or

magma: E := EllipticCurve("84150j1");

sage: E = EllipticCurve([1, -1, 0, -25542, 2802116]) # or

sage: E = EllipticCurve("84150j1")

gp: E = ellinit([1, -1, 0, -25542, 2802116]) \\ or

gp: E = ellinit("84150j1")

$$y^2 + x y = x^{3} - x^{2} - 25542 x + 2802116$$

Mordell-Weil group structure

$$\Z^2$$

Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-71, 2098\right)$$ $$\left(-176, 1438\right)$$ $$\hat{h}(P)$$ ≈ 0.229535290808 2.83608937001

Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-196, 598\right)$$, $$\left(-176, 1438\right)$$, $$\left(-115, 2109\right)$$, $$\left(-71, 2098\right)$$, $$\left(28, 1438\right)$$, $$\left(149, 1438\right)$$, $$\left(424, 8038\right)$$, $$\left(688, 17278\right)$$, $$\left(1304, 46098\right)$$, $$\left(2179, 100348\right)$$, $$\left(8929, 839098\right)$$

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

Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$84150$$ = $$2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \cdot 17$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-2310075281250000$$ = $$-1 \cdot 2^{4} \cdot 3^{3} \cdot 5^{9} \cdot 11^{5} \cdot 17$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{4368317413923}{5475734000}$$ = $$-1 \cdot 2^{-4} \cdot 3^{3} \cdot 5^{-3} \cdot 11^{-5} \cdot 17^{-1} \cdot 5449^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$2$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$0.24564178367$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.416268775683$$ 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: $$80$$  = $$2\cdot2\cdot2^{2}\cdot5\cdot1$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$1$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (rounded)

Modular invariants

Modular form 84150.2.a.bd

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)

$$q - q^{2} + q^{4} - q^{7} - q^{8} + q^{11} - q^{13} + q^{14} + q^{16} - q^{17} - 7q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 460800 $$\Gamma_0(N)$$-optimal: yes 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])

$$L^{(2)}(E,1)/2!$$ ≈ $$8.18024036358$$

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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$3$$ $$2$$ $$III$$ Additive 1 2 3 0
$$5$$ $$4$$ $$I_3^{*}$$ Additive 1 2 9 3
$$11$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$17$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1

Galois representations

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

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 $$p$$ .

$p$-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 add add ordinary split ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss 3 - - 2 3 2 2 2 2 2 2 2 2 2 2,2 0 - - 0 0 0 0 0 0 0 0 0 0 0 0,0

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has no rational isogenies. Its isogeny class 84150.bd consists of this curve only.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

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