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## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -2968494, 1967687156])

gp: E = ellinit([1, 1, 0, -2968494, 1967687156])

magma: E := EllipticCurve([1, 1, 0, -2968494, 1967687156]);

$$y^2+xy=x^3+x^2-2968494x+1967687156$$ ## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(1004, 290\right)$$ $$\hat{h}(P)$$ ≈ $4.5397035579255599921322872330$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(1004, 290\right)$$, $$\left(1004, -1294\right)$$ ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$51842$$ = $$2 \cdot 7^{2} \cdot 23^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-589645382868759616$$ = $$-1 \cdot 2^{6} \cdot 7^{6} \cdot 23^{8}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{313994137}{64}$$ = $$-1 \cdot 2^{-6} \cdot 23 \cdot 239^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$2.4073818182010743069563533048\dots$$ Stable Faltings height: $$-0.65590273361268213946749162146\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$4.5397035579255599921322872330\dots$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$0.28201650825558199921644838496\dots$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$4$$  = $$2\cdot2\cdot1$$ 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 51842.2.a.i

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - q^{2} + 2q^{3} + q^{4} - 3q^{5} - 2q^{6} - q^{8} + q^{9} + 3q^{10} + 6q^{11} + 2q^{12} + q^{13} - 6q^{15} + q^{16} - 6q^{17} - q^{18} - 2q^{19} + O(q^{20})$$ For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1987200 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$5.1210853836864346564411741552048611064$$

## Local data

This elliptic curve is not semistable. There are 3 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{6}$$ Non-split multiplicative 1 1 6 6
$$7$$ $$2$$ $$I_0^{*}$$ Additive -1 2 6 0
$$23$$ $$1$$ $$IV^{*}$$ Additive -1 2 8 0

## 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 X3.

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

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 nonsplit ordinary ordinary add ordinary ordinary ordinary ordinary add ordinary ordinary ordinary ordinary ordinary ordinary 6 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

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 51842.i consists of 2 curves linked by isogenies of degree 3.

## 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 $K$ $2$ $$\Q(\sqrt{21})$$ $$\Z/3\Z$$ Not in database $3$ 3.1.2116.1 $$\Z/2\Z$$ Not in database $6$ 6.0.17909824.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.0.69973402527.1 $$\Z/3\Z$$ Not in database $6$ 6.2.41465720016.3 $$\Z/6\Z$$ 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 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.6.27621688148894336546280790907368951762944.1 $$\Z/9\Z$$ Not in database $18$ 18.0.1403327142655811438616104806552301568.1 $$\Z/6\Z$$ 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.