# Properties

 Label 21780x1 Conductor $21780$ Discriminant $-1.253\times 10^{19}$ j-invariant $$\frac{72268906496}{606436875}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, 238128, -164337239])

gp: E = ellinit([0, 0, 0, 238128, -164337239])

magma: E := EllipticCurve([0, 0, 0, 238128, -164337239]);

$$y^2=x^3+238128x-164337239$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(462, 6655\right)$$ $$\hat{h}(P)$$ ≈ $0.94212873102341232877614496096$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(407, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(407, 0\right)$$, $$(462,\pm 6655)$$, $$(650,\pm 16281)$$, $$(3432,\pm 202675)$$, $$(13772,\pm 1617165)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$21780$$ = $$2^{2} \cdot 3^{2} \cdot 5 \cdot 11^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-12531100788527310000$$ = $$-1 \cdot 2^{4} \cdot 3^{12} \cdot 5^{4} \cdot 11^{9}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{72268906496}{606436875}$$ = $$2^{20} \cdot 3^{-6} \cdot 5^{-4} \cdot 11^{-3} \cdot 41^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$2.3459282201452818573870739533\dots$$ Stable Faltings height: $$0.36662537922539330318606883870\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.94212873102341232877614496096\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.11150142807795659595832832673\dots$$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $$192$$  = $$3\cdot2^{2}\cdot2^{2}\cdot2^{2}$$ sage: E.torsion_order()  gp: elltors(E)[1]  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 21780.2.a.r

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

magma: ModularForm(E);

$$q + q^{5} - 2q^{7} - 2q^{13} - 2q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 276480 $$\Gamma_0(N)$$-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[2]/factorial(ar[1])

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

$$L'(E,1)$$ ≈ $$5.0423375492344091959905790215277806986$$

## Local data

This elliptic curve is not semistable. There are 4 primes of bad reduction:

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$3$$ $$IV$$ Additive -1 2 4 0
$$3$$ $$4$$ $$I_6^{*}$$ Additive -1 2 12 6
$$5$$ $$4$$ $$I_{4}$$ Split multiplicative -1 1 4 4
$$11$$ $$4$$ $$I_3^{*}$$ Additive -1 2 9 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 X6.

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

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(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 add split ordinary add ordinary ss ordinary ss ss ordinary ordinary ss ordinary ss - - 2 3 - 1 1,1 1 1,1 1,1 3 1 1,1 1 1,1 - - 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 non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 21780x consists of 4 curves linked by isogenies of degrees dividing 6.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-11})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{33})$$ $$\Z/6\Z$$ Not in database $4$ 4.2.6336.1 $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ $$\Z/2\Z \times \Z/6\Z$$ Not in database $6$ 6.0.119790000.2 $$\Z/2\Z \times \Z/6\Z$$ Not in database $8$ 8.0.587761422336.7 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.4857532416.7 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.4.4857532416.1 $$\Z/12\Z$$ Not in database $12$ Deg 12 $$\Z/6\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ Not in database $16$ 16.0.23595621172490797056.2 $$\Z/2\Z \times \Z/12\Z$$ Not in database $18$ 18.6.37154520920503664275418810704756800000000.1 $$\Z/18\Z$$ 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.