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## Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -1486, 11789])

gp: E = ellinit([0, -1, 0, -1486, 11789])

magma: E := EllipticCurve([0, -1, 0, -1486, 11789]);

$$y^2=x^3-x^2-1486x+11789$$ ## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(7, 41\right)$$ $$\hat{h}(P)$$ ≈ $1.7480782010275749248215930729$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(7,\pm 41)$$ ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$8036$$ = $$2^{2} \cdot 7^{2} \cdot 41$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$155050087696$$ = $$2^{4} \cdot 7^{8} \cdot 41^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{3937024}{1681}$$ = $$2^{8} \cdot 7 \cdot 13^{3} \cdot 41^{-2}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$0.84380288168284167315809491062\dots$$ Stable Faltings height: $$-0.68451961120734896671788429216\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1.7480782010275749248215930729\dots$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$0.92590731154269220563151130529\dots$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$2$$  = $$1\cdot1\cdot2$$ 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

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - q^{3} - q^{5} - 2q^{9} + 3q^{11} + 2q^{13} + q^{15} + q^{17} - 3q^{19} + O(q^{20})$$ sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 4032 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$3.2371167749596555000492148902203259411$$

## 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$$(j)_{-}$$
$$2$$ $$1$$ $$IV$$ Additive -1 2 4 0
$$7$$ $$1$$ $$IV^{*}$$ Additive 1 2 8 0
$$41$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2

## 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 X2.

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

## $p$-adic data

### $p$-adic regulators

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

Note: $$p$$-adic regulator data only exists for primes $$p\ge 5$$ of good ordinary reduction.

## 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 add ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary split ordinary ordinary - 1 1 - 1 1 1 1 1 1 1 1 2 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 no rational isogenies. Its isogeny class 8036c 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 $K$ $3$ $$\Q(\zeta_{7})^+$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ 8.2.237408548417712.4 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/4\Z$$ 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.