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## Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -2542800, 1493548992]) # or

sage: E = EllipticCurve("21840bk3")

gp: E = ellinit([0, -1, 0, -2542800, 1493548992]) \\ or

gp: E = ellinit("21840bk3")

magma: E := EllipticCurve([0, -1, 0, -2542800, 1493548992]); // or

magma: E := EllipticCurve("21840bk3");

$$y^2 = x^{3} - x^{2} - 2542800 x + 1493548992$$

## Mordell-Weil group structure

$$\Z/{2}\Z$$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-1832, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-1832, 0\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor: $$21840$$ = $$2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$89680698351738224640$$ = $$2^{16} \cdot 3^{2} \cdot 5 \cdot 7^{12} \cdot 13^{3}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{443915739051786565201}{21894701746029840}$$ = $$2^{-4} \cdot 3^{-2} \cdot 5^{-1} \cdot 7^{-12} \cdot 11^{3} \cdot 13^{-3} \cdot 37^{3} \cdot 18743^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$0$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$1$$ sage: E.period_lattice().omega()  gp: E.omega  magma: RealPeriod(E); Real period: $$0.188556931634$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[i,1],gr[i]] | i<-[1..#gr[,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$48$$  = $$2^{2}\cdot2\cdot1\cdot2\cdot3$$ sage: E.torsion_order()  gp: elltors(E)  magma: Order(TorsionSubgroup(E)); Torsion order: $$2$$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 21840.2.a.w

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy)/(2*xy+E.a1*xy+E.a3)

magma: ModularForm(E);

$$q - q^{3} + q^{5} - q^{7} + q^{9} + q^{13} - q^{15} + 6q^{17} + 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 663552 $$\Gamma_0(N)$$-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);

$$L(E,1)$$ ≈ $$2.26268317961$$

## Local data

This elliptic curve is not semistable.

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$$(j)_{-}$$
$$2$$ $$4$$ $$I_8^{*}$$ Additive -1 4 16 4
$$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$5$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$7$$ $$2$$ $$I_{12}$$ Non-split multiplicative 1 1 12 12
$$13$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3

## 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 X13f.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 3 \\ 0 & 1 \end{array}\right)$ and has index 12.

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
$$2$$ B
$$3$$ B

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) 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 13 add nonsplit split nonsplit split - 2 1 0 1 - 0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

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 $$d=$$ 2, 3, 4, 6 and 12.
Its isogeny class 21840.w consists of 8 curves linked by isogenies of degrees dividing 12.

## Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{-1})$$ $$\Z/4\Z$$ Not in database
$$\Q(\sqrt{3})$$ $$\Z/6\Z$$ Not in database
$$\Q(\sqrt{65})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
$$\Q(\sqrt{-65})$$ $$\Z/4\Z$$ Not in database
4 $$\Q(\zeta_{12})$$ $$\Z/12\Z$$ Not in database
$$\Q(i, \sqrt{65})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database
$$\Q(\sqrt{3}, \sqrt{-65})$$ $$\Z/12\Z$$ Not in database
$$\Q(\sqrt{3}, \sqrt{65})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database
4.2.9886500.2 $$\Z/2\Z \times \Z/4\Z$$ Not in database
6 6.0.9447840000.12 $$\Z/12\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.