# Properties

 Label 84150j1 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()

$$\left(-196, 598\right)$$, $$\left(-196, -402\right)$$, $$\left(-176, 1438\right)$$, $$\left(-176, -1262\right)$$, $$\left(-115, 2109\right)$$, $$\left(-115, -1994\right)$$, $$\left(-71, 2098\right)$$, $$\left(-71, -2027\right)$$, $$\left(28, 1438\right)$$, $$\left(28, -1466\right)$$, $$\left(149, 1438\right)$$, $$\left(149, -1587\right)$$, $$\left(424, 8038\right)$$, $$\left(424, -8462\right)$$, $$\left(688, 17278\right)$$, $$\left(688, -17966\right)$$, $$\left(1304, 46098\right)$$, $$\left(1304, -47402\right)$$, $$\left(2179, 100348\right)$$, $$\left(2179, -102527\right)$$, $$\left(8929, 839098\right)$$, $$\left(8929, -848027\right)$$

## 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 84150j 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.