# Properties

 Label 88445bn3 Conductor $88445$ Discriminant $-7.567\times 10^{19}$ j-invariant $$-\frac{250523582464}{13671875}$$ CM no Rank $2$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 1, -2323155, -1426494191])

gp: E = ellinit([0, 1, 1, -2323155, -1426494191])

magma: E := EllipticCurve([0, 1, 1, -2323155, -1426494191]);

$$y^2+y=x^3+x^2-2323155x-1426494191$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(3901, 221112\right)$$ $$\left(\frac{17959}{9}, \frac{1172924}{27}\right)$$ $$\hat{h}(P)$$ ≈ $0.59373624833574598489644686192$ $4.5130786330362349497326755551$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(1801, 15312\right)$$, $$\left(1801, -15313\right)$$, $$\left(3901, 221112\right)$$, $$\left(3901, -221113\right)$$, $$\left(17131, 2233052\right)$$, $$\left(17131, -2233053\right)$$, $$\left(54961, 12880032\right)$$, $$\left(54961, -12880033\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$88445$$ = $$5 \cdot 7^{2} \cdot 19^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-75672472610123046875$$ = $$-1 \cdot 5^{9} \cdot 7^{7} \cdot 19^{6}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{250523582464}{13671875}$$ = $$-1 \cdot 2^{15} \cdot 5^{-9} \cdot 7^{-1} \cdot 197^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$2.5726366877418816168648579893\dots$$ Stable Faltings height: $$0.12746212363100473430766790164\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$2.6229490984078692854595441562\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.060950134314442934087428220119\dots$$ 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: $$72$$  = $$3^{2}\cdot2^{2}\cdot2$$ 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: $$1$$ (rounded)

## Modular invariants

Modular form 88445.2.a.w

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2052864 $$\Gamma_0(N)$$-optimal: no 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);

$$L^{(2)}(E,1)/2!$$ ≈ $$11.510575189049277364938363780651675809$$

## Local data

This elliptic curve is not semistable. There are 3 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$$(j)_{-}$$
$$5$$ $$9$$ $$I_{9}$$ Split multiplicative -1 1 9 9
$$7$$ $$4$$ $$I_1^{*}$$ Additive -1 2 7 1
$$19$$ $$2$$ $$I_0^{*}$$ Additive -1 2 6 0

## 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
$$3$$ B

## $p$-adic data

### $p$-adic regulators

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

$$p$$-adic regulators are not yet computed for curves that are not $$\Gamma_0$$-optimal.

## 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 ordinary split add ordinary ordinary ordinary add ordinary ordinary ordinary ordinary ordinary ordinary ordinary 4,9 2 3 - 2 2 2 - 4 2 2 2 2 2 2 0,0 2 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 $$d=$$ 3 and 9.
Its isogeny class 88445bn consists of 3 curves linked by isogenies of degrees dividing 9.

## 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 $K$ $2$ $$\Q(\sqrt{-399})$$ $$\Z/3\Z$$ Not in database $3$ 3.1.140.1 $$\Z/2\Z$$ Not in database $6$ 6.0.686000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.2.84038546277.1 $$\Z/3\Z$$ Not in database $6$ 6.0.2269040749479.7 $$\Z/9\Z$$ Not in database $6$ 6.0.25408479600.5 $$\Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ Deg 12 $$\Z/3\Z \times \Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/9\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.2.37985300612755663104180941142779712000000.1 $$\Z/6\Z$$ Not in database $18$ 18.0.747664671960869716879593464513333071296000000.2 $$\Z/18\Z$$ 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.