# Properties

 Label 2205.e3 Conductor $2205$ Discriminant $-3.677\times 10^{12}$ j-invariant $$\frac{71991296}{42875}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 1, 3822, -15521])

gp: E = ellinit([0, 0, 1, 3822, -15521])

magma: E := EllipticCurve([0, 0, 1, 3822, -15521]);

$$y^2+y=x^3+3822x-15521$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(77, 857\right)$$ $$\hat{h}(P)$$ ≈ $1.3523886126642278844925932276$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(77, 857\right)$$, $$\left(77, -858\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$2205$$ = $$3^{2} \cdot 5 \cdot 7^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-3677222437875$$ = $$-1 \cdot 3^{6} \cdot 5^{3} \cdot 7^{9}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{71991296}{42875}$$ = $$2^{15} \cdot 5^{-3} \cdot 7^{-3} \cdot 13^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.1004171981586613868603442734\dots$$ Stable Faltings height: $$-0.42184402070305011138995471678\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1.3523886126642278844925932276\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.46016342288155861202787157474\dots$$ 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: $$4$$  = $$1\cdot1\cdot2^{2}$$ 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);

$$q - 2q^{4} - q^{5} + 3q^{11} - 5q^{13} + 4q^{16} + 3q^{17} - 2q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2880 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$2.4892790922784538744740319578205463377$$

## 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$$(j)_{-}$$
$$3$$ $$1$$ $$I_0^{*}$$ Additive -1 2 6 0
$$5$$ $$1$$ $$I_{3}$$ Non-split multiplicative 1 1 3 3
$$7$$ $$4$$ $$I_3^{*}$$ Additive -1 2 9 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
$$3$$ Cs

## $p$-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 add nonsplit add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary 2,7 - 1 - 1 1 1 1 1 1 1 1 1 1 1 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 $$d=$$ 3.
Its isogeny class 2205.e consists of 2 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 $K$ $2$ $$\Q(\sqrt{21})$$ $$\Z/3\Z$$ 2.2.21.1-175.1-a3 $2$ $$\Q(\sqrt{-7})$$ $$\Z/3\Z$$ 2.0.7.1-14175.1-b3 $3$ 3.1.140.1 $$\Z/2\Z$$ Not in database $4$ $$\Q(\sqrt{-3}, \sqrt{-7})$$ $$\Z/3\Z \times \Z/3\Z$$ Not in database $6$ 6.0.686000.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.2.3704400.1 $$\Z/6\Z$$ Not in database $6$ 6.0.137200.1 $$\Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ 12.0.13722579360000.1 $$\Z/3\Z \times \Z/6\Z$$ Not in database $12$ 12.0.343064484000000.3 $$\Z/2\Z \times \Z/6\Z$$ Not in database $12$ 12.0.470596000000.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.6.49661170246971776397429.2 $$\Z/9\Z$$ Not in database $18$ 18.0.2946207121585483804827941162109375.2 $$\Z/9\Z$$ 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.