Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -324333438, 2248285015576])

gp: E = ellinit([1, -1, 1, -324333438, 2248285015576])

magma: E := EllipticCurve([1, -1, 1, -324333438, 2248285015576]);

$$y^2+xy+y=x^3-x^2-324333438x+2248285015576$$ trivial

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E); ## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$18515$$ = $$5 \cdot 7 \cdot 23^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-15800653619061292835$$ = $$-1 \cdot 5 \cdot 7^{9} \cdot 23^{8}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{48181043296511332209}{201768035}$$ = $$-1 \cdot 3^{3} \cdot 5^{-1} \cdot 7^{-9} \cdot 23 \cdot 97^{3} \cdot 4397^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$3.3153242848026272473887412595\dots$$ Stable Faltings height: $$1.2249948075165274535175727050\dots$$

## 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: $$0.14833863653993797501804721595\dots$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$1$$ 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: $$16$$ = $4^2$ (exact)

## Modular invariants

Modular form 18515.2.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);

$$q - q^{2} + 3q^{3} - q^{4} - q^{5} - 3q^{6} - q^{7} + 3q^{8} + 6q^{9} + q^{10} - q^{11} - 3q^{12} - 2q^{13} + q^{14} - 3q^{15} - q^{16} + 3q^{17} - 6q^{18} + 4q^{19} + O(q^{20})$$ For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 4292352 $$\Gamma_0(N)$$-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);

$$L(E,1)$$ ≈ $$2.3734181846390076002887554551546877411$$

## 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$$(j)_{-}$$
$$5$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$7$$ $$1$$ $$I_{9}$$ Non-split multiplicative 1 1 9 9
$$23$$ $$1$$ $$IV^{*}$$ Additive -1 2 8 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 $$p$$ .

## $p$-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 ordinary ss nonsplit nonsplit ordinary ordinary ordinary ordinary add ordinary ordinary ordinary ordinary ss ordinary 3 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 0,0 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 18515.i 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 $K$ $3$ 3.1.18515.1 $$\Z/2\Z$$ Not in database $6$ 6.0.11998182875.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ 8.2.382507666875.2 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ 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.