# Properties

 Label 8085j1 Conductor $8085$ Discriminant $-1.999\times 10^{18}$ j-invariant $$-\frac{79028701534867456}{16987307596875}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 1, -438076, -130549434])

gp: E = ellinit([0, -1, 1, -438076, -130549434])

magma: E := EllipticCurve([0, -1, 1, -438076, -130549434]);

$$y^2+y=x^3-x^2-438076x-130549434$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(810, 6737\right)$$ $$\hat{h}(P)$$ ≈ $3.5503903399671316327844895691$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(810, 6737\right)$$, $$\left(810, -6738\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$8085$$ = $$3 \cdot 5 \cdot 7^{2} \cdot 11$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-1998539751464746875$$ = $$-1 \cdot 3^{5} \cdot 5^{5} \cdot 7^{11} \cdot 11^{3}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{79028701534867456}{16987307596875}$$ = $$-1 \cdot 2^{12} \cdot 3^{-5} \cdot 5^{-5} \cdot 7^{-5} \cdot 11^{-3} \cdot 26821^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$2.2327201502027413440576191766\dots$$ Stable Faltings height: $$1.2597650756750846915049428049\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$3.5503903399671316327844895691\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.091731218898426550770618201899\dots$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$6$$  = $$1\cdot1\cdot2\cdot3$$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $$1$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

magma: ModularForm(E);

$$q - 2q^{2} - q^{3} + 2q^{4} - q^{5} + 2q^{6} + q^{9} + 2q^{10} + q^{11} - 2q^{12} + 6q^{13} + q^{15} - 4q^{16} + 7q^{17} - 2q^{18} + 5q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 288000 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar[2]/factorial(ar[1])

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

$$L'(E,1)$$ ≈ $$1.9540898007023040700058553944641308119$$

## Local data

This elliptic curve is not semistable. There are 4 primes of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$3$$ $$1$$ $$I_{5}$$ Non-split multiplicative 1 1 5 5
$$5$$ $$1$$ $$I_{5}$$ Non-split multiplicative 1 1 5 5
$$7$$ $$2$$ $$I_5^{*}$$ Additive -1 2 11 5
$$11$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3

## Galois representations

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

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

The mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$5$$ B.4.1

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

Note: $$p$$-adic regulator data only exists for primes $$p\ge 5$$ 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 ss nonsplit nonsplit add split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary 10,3 1 1 - 2 1 1 1 1 1 1 1 1 1 1 0,0 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 non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 5.
Its isogeny class 8085j consists of 2 curves linked by isogenies of degree 5.

## Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-7})$$ $$\Z/5\Z$$ Not in database $3$ 3.1.4620.1 $$\Z/2\Z$$ Not in database $6$ 6.0.24652782000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.0.149410800.1 $$\Z/10\Z$$ Not in database $8$ 8.2.13025729626875.2 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/10\Z$$ Not in database $16$ Deg 16 $$\Z/15\Z$$ Not in database $20$ 20.4.396107830343483954099825714111328125.1 $$\Z/5\Z$$ Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.