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## Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, 963, -829])

gp: E = ellinit([0, -1, 0, 963, -829])

magma: E := EllipticCurve([0, -1, 0, 963, -829]);

$$y^2=x^3-x^2+963x-829$$ ## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(86, 843\right)$$ $$\hat{h}(P)$$ ≈ $4.6321794344324875663979184694$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(86,\pm 843)$$ ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$23104$$ = $$2^{6} \cdot 19^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-57207791296$$ = $$-1 \cdot 2^{6} \cdot 19^{7}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{32768}{19}$$ = $$2^{15} \cdot 19^{-1}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$0.75361994818640099683797942334\dots$$ Stable Faltings height: $$-1.0651731316767918878751503533\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$4.6321794344324875663979184694\dots$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$0.66174646908858210195637988201\dots$$ 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 23104.2.a.bs

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q + 2q^{3} - 3q^{5} + q^{7} + q^{9} + 3q^{11} - 4q^{13} - 6q^{15} - 3q^{17} + O(q^{20})$$ sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 17280 $$\Gamma_0(N)$$-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);

$$L'(E,1)$$ ≈ $$6.1306567698408877137827406166205018627$$

## 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 6 6 0
$$19$$ $$2$$ $$I_1^{*}$$ Additive -1 2 7 1

## 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]

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 ordinary ordinary ordinary ordinary add ss ordinary ordinary ordinary ordinary ordinary ordinary - 1 1 1 1 1 1 - 1,1 3 1 1 1 1 1 - 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 non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3 and 9.
Its isogeny class 23104bz 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 $K$ $2$ $$\Q(\sqrt{38})$$ $$\Z/3\Z$$ 2.2.152.1-19.1-b3 $3$ 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 $6$ 6.0.34229592576.3 $$\Z/3\Z$$ Not in database $6$ 6.6.1267762688.1 $$\Z/9\Z$$ Not in database $6$ 6.2.14047232.3 $$\Z/6\Z$$ Not in database $12$ 12.2.59986716965994496.21 $$\Z/4\Z$$ 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 $12$ 12.0.197324726861824.14 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.0.2566759414823783670846037073854464.4 $$\Z/6\Z$$ Not in database $18$ 18.6.130404888219467747337602859008.1 $$\Z/18\Z$$ 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.