# Properties

 Label 214774.o4 Conductor $214774$ Discriminant $-1.598\times 10^{22}$ j-invariant $$-\frac{10090256344188054273}{107965577101312}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -23814886, -45137954955])

gp: E = ellinit([1, -1, 1, -23814886, -45137954955])

magma: E := EllipticCurve([1, -1, 1, -23814886, -45137954955]);

$$y^2+xy+y=x^3-x^2-23814886x-45137954955$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(254409, 128170107\right)$$ $\hat{h}(P)$ ≈ $6.7229321327543385393184602332$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(5641, -2821\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(5641, -2821\right)$$, $$\left(254409, 128170107\right)$$, $$\left(254409, -128424517\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$214774$$ = $2 \cdot 7 \cdot 23^{2} \cdot 29$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-15982780187590764986368$ = $-1 \cdot 2^{16} \cdot 7 \cdot 23^{10} \cdot 29^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{10090256344188054273}{107965577101312}$$ = $-1 \cdot 2^{-16} \cdot 3^{3} \cdot 7^{-1} \cdot 23^{-4} \cdot 29^{-2} \cdot 720299^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $3.0768212499328656427738452548\dots$ Stable Faltings height: $1.5090741419682907973704688389\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $6.7229321327543385393184602332\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.034149196260523650848143329428\dots$ 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: $128$  = $2^{4}\cdot1\cdot2^{2}\cdot2$ 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) 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); Special value: $L'(E,1)$ ≈ $7.3466473231234800082878296677752085953$

## Modular invariants

Modular form 214774.2.a.o

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);

$$q + q^{2} + q^{4} - 2 q^{5} + q^{7} + q^{8} - 3 q^{9} - 2 q^{10} - 4 q^{11} + 6 q^{13} + q^{14} + q^{16} - 6 q^{17} - 3 q^{18} - 4 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 20275200 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## 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$(j)_{-}$
$2$ $16$ $I_{16}$ Split multiplicative -1 1 16 16
$7$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$23$ $4$ $I_{4}^{*}$ Additive -1 2 10 4
$29$ $2$ $I_{2}$ Split multiplicative -1 1 2 2

## Galois representations

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 $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 8.12.0.6

## $p$-adic regulators

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

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.

No Iwasawa invariant data is available for this curve.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 214774.o consists of 4 curves linked by isogenies of degrees dividing 4.

## 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 $K$ $2$ $$\Q(\sqrt{-7})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{161})$$ $$\Z/4\Z$$ Not in database $2$ $$\Q(\sqrt{-23})$$ $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-7}, \sqrt{-23})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ 4.2.152596927.1 $$\Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.3510325504.1 $$\Z/8\Z$$ Not in database $8$ 8.0.23285822129843329.11 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/4\Z \times \Z/4\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/12\Z$$ 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.