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## Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 1, -1293249, -565640702])

gp: E = ellinit([0, -1, 1, -1293249, -565640702])

magma: E := EllipticCurve([0, -1, 1, -1293249, -565640702]);

$$y^2+y=x^3-x^2-1293249x-565640702$$ ## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(\frac{4490897913973861300437037109}{3078549237716740290520900}, \frac{137706327894684946735684190677072440042127}{5401559396184247367428848977609677000}\right)$$ $$\hat{h}(P)$$ ≈ $60.329741894448769023784325916$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E); ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$31939$$ = $$19 \cdot 41^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-90251980579$$ = $$-1 \cdot 19 \cdot 41^{6}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{50357871050752}{19}$$ = $$-1 \cdot 2^{18} \cdot 19^{-1} \cdot 577^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.8902251903434717054534765701\dots$$ Stable Faltings height: $$0.033439156991317803520094883582\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$60.329741894448769023784325916\dots$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$0.070786264281172631894277233195\dots$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$2$$  = $$1\cdot2$$ 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 31939.2.a.e

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q + 2q^{3} - 2q^{4} + 3q^{5} + q^{7} + q^{9} - 3q^{11} - 4q^{12} + 4q^{13} + 6q^{15} + 4q^{16} + 3q^{17} - q^{19} + O(q^{20})$$ For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 207360 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$8.5410341075107660171144378303695275330$$

## 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$$(j)_{-}$$
$$19$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$41$$ $$2$$ $$I_0^{*}$$ 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 ordinary ordinary ordinary ordinary ordinary nonsplit ss ordinary ordinary ordinary add ordinary ordinary 4,7 1 3 1 3 3 1 1 1,1 1 1 1 - 1 1 0,0 2 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 and 9.
Its isogeny class 31939.e 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 $K$ $2$ $$\Q(\sqrt{-123})$$ $$\Z/3\Z$$ Not in database $3$ 3.1.76.1 $$\Z/2\Z$$ Not in database $6$ 6.0.109744.2 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.2.80836682769.1 $$\Z/3\Z$$ Not in database $6$ 6.0.176789825215803.1 $$\Z/9\Z$$ Not in database $6$ 6.0.10748367792.4 $$\Z/6\Z$$ 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 $18$ 18.2.781074752908860659443693246733933973504.2 $$\Z/6\Z$$ Not in database $18$ 18.0.8170317793372634166092528851622775528533083533312.1 $$\Z/18\Z$$ 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.