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## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, 2781539, 4194361253])

gp: E = ellinit([1, -1, 1, 2781539, 4194361253])

magma: E := EllipticCurve([1, -1, 1, 2781539, 4194361253]);

$$y^2+xy+y=x^3-x^2+2781539x+4194361253$$ trivial

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E); ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$54418$$ = $$2 \cdot 7 \cdot 13^{2} \cdot 23$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-8979874391155699892224$$ = $$-1 \cdot 2^{14} \cdot 7^{4} \cdot 13^{8} \cdot 23^{4}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{2917645350022143}{11008380780544}$$ = $$2^{-14} \cdot 3^{3} \cdot 7^{-4} \cdot 13 \cdot 23^{-4} \cdot 47^{3} \cdot 431^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$2.8952532001045967297893622010\dots$$ Stable Faltings height: $$1.1852869617969055724203705733\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$0$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$0.092535650103349596531327854425\dots$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$56$$  = $$( 2 \cdot 7 )\cdot2\cdot1\cdot2$$ 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

Modular form 54418.2.a.f

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q + q^{2} + q^{4} + 3q^{5} - q^{7} + q^{8} - 3q^{9} + 3q^{10} - q^{14} + q^{16} + 3q^{17} - 3q^{18} + 8q^{19} + O(q^{20})$$ For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 3983616 $$\Gamma_0(N)$$-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/factorial(ar)

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

$$L(E,1)$$ ≈ $$5.1819964057875774057543598477698769924$$

## Local data

This elliptic curve is not semistable. There are 4 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$$(j)_{-}$$
$$2$$ $$14$$ $$I_{14}$$ Split multiplicative -1 1 14 14
$$7$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$13$$ $$1$$ $$IV^{*}$$ Additive 1 2 8 0
$$23$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4

## Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X20b.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 3 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 3 \\ 1 & 0 \end{array}\right)$ and has index 16.

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 $$p$$ .

## $p$-adic data

### $p$-adic regulators

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

All $$p$$-adic regulators are identically $$1$$ since the rank is $$0$$.

## 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 split ss ordinary nonsplit ss add ordinary ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ss 10 0,0 0 4 0,0 - 0 0 0 0 0 0 0 0 0,0 0 0,0 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 no rational isogenies. Its isogeny class 54418.f consists of this curve only.

## 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 $K$ $3$ 3.1.676.1 $$\Z/4\Z$$ Not in database $6$ 6.0.1827904.2 $$\Z/4\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/8\Z$$ 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.