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## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -53909, -5143421])

gp: E = ellinit([0, 1, 0, -53909, -5143421])

magma: E := EllipticCurve([0, 1, 0, -53909, -5143421]);

$$y^2=x^3+x^2-53909x-5143421$$ ## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(\frac{29309}{100}, \frac{2078277}{1000}\right)$$ $$\hat{h}(P)$$ ≈ $5.5146269853974668340729678201$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E); ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$5776$$ = $$2^{4} \cdot 19^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-1321728810102784$$ = $$-1 \cdot 2^{12} \cdot 19^{9}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{89915392}{6859}$$ = $$-1 \cdot 2^{18} \cdot 7^{3} \cdot 19^{-3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.6494996828004284972442181025\dots$$ Stable Faltings height: $$-0.51586698734273704217752773490\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$5.5146269853974668340729678201\dots$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$0.15597513857293015318602148328\dots$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$4$$  = $$1\cdot2^{2}$$ sage: E.torsion_order()  gp: elltors(E)  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)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - 2q^{3} + 3q^{5} + q^{7} + q^{9} - 3q^{11} + 4q^{13} - 6q^{15} - 3q^{17} + O(q^{20})$$ For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 25920 $$\Gamma_0(N)$$-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/factorial(ar)

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

$$L'(E,1)$$ ≈ $$3.4405788329015598312099711537409075025$$

## Local data

This elliptic curve is not semistable. There are 2 primes of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)

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

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$1$$ $$II^{*}$$ Additive -1 4 12 0
$$19$$ $$4$$ $$I_3^{*}$$ 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 add ordinary ordinary ordinary ordinary ordinary ordinary add ss ordinary ordinary ordinary ordinary ordinary ordinary - 3 3 1 3 1 1 - 1,1 1 1 1 1 1 1 - 1 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 5776q 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 $K$ $2$ $$\Q(\sqrt{19})$$ $$\Z/3\Z$$ 2.2.76.1-19.1-b2 $2$ $$\Q(\sqrt{-57})$$ $$\Z/3\Z$$ Not in database $3$ 3.1.76.1 $$\Z/2\Z$$ Not in database $4$ $$\Q(\sqrt{-3}, \sqrt{19})$$ $$\Z/3\Z \times \Z/3\Z$$ Not in database $6$ 6.0.109744.2 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.2.1755904.2 $$\Z/6\Z$$ Not in database $6$ 6.0.47409408.1 $$\Z/6\Z$$ Not in database $12$ 12.2.937292452593664.5 $$\Z/4\Z$$ Not in database $12$ 12.0.2247651966910464.2 $$\Z/3\Z \times \Z/6\Z$$ Not in database $12$ 12.0.3083198857216.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database $12$ 12.0.2247651966910464.3 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.6.2901158554443114862338637824.1 $$\Z/9\Z$$ Not in database $18$ 18.0.30347143177379886431560874232935350272.4 $$\Z/9\Z$$ Not in database

We only show fields where the torsion growth is primitive.