# Properties

 Label 212160go1 Conductor $212160$ Discriminant $2.722\times 10^{17}$ j-invariant $$\frac{2396726313900986596}{4154072495625}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3-x^2-1124065x+458394337$$ y^2=x^3-x^2-1124065x+458394337 (homogenize, simplify) $$y^2z=x^3-x^2z-1124065xz^2+458394337z^3$$ y^2z=x^3-x^2z-1124065xz^2+458394337z^3 (dehomogenize, simplify) $$y^2=x^3-91049292x+333896323824$$ y^2=x^3-91049292x+333896323824 (homogenize, minimize)

sage: E = EllipticCurve([0, -1, 0, -1124065, 458394337])

gp: E = ellinit([0, -1, 0, -1124065, 458394337])

magma: E := EllipticCurve([0, -1, 0, -1124065, 458394337]);

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(424, 7605\right)$$ (424, 7605) $\hat{h}(P)$ ≈ $0.49382041103376956989013427096$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(593, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-1071,\pm 20800)$$, $$(64,\pm 19665)$$, $$(424,\pm 7605)$$, $$\left(593, 0\right)$$, $$(1009,\pm 18720)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$212160$$ = $2^{6} \cdot 3 \cdot 5 \cdot 13 \cdot 17$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $272241295073280000$ = $2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 13^{6} \cdot 17$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{2396726313900986596}{4154072495625}$$ = $2^{2} \cdot 3^{-4} \cdot 5^{-4} \cdot 13^{-6} \cdot 17^{-1} \cdot 19^{3} \cdot 44371^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.2393375922581837715701781561\dots$ Stable Faltings height: $1.3151413515115900256805353275\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $0.49382041103376956989013427096\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.30956310408703058585904038157\dots$ 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: $192$  = $2^{2}\cdot2\cdot2^{2}\cdot( 2 \cdot 3 )\cdot1$ 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: $L'(E,1)$ ≈ $7.3376918064550577547530879346$

## Modular invariants

Modular form 212160.2.a.cy

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);

$$q - q^{3} + q^{5} + q^{9} - 2 q^{11} + q^{13} - q^{15} - q^{17} + 8 q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 2949120 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 5 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$(j)_{-}$
$2$ $4$ $I_{6}^{*}$ Additive 1 6 16 0
$3$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$5$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$13$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$17$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

## 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 2.3.0.1

## $p$-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.
Its isogeny class 212160go consists of 2 curves linked by isogenies of degree 2.

## 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 $K$ $2$ $$\Q(\sqrt{17})$$ $$\Z/2\Z \oplus \Z/2\Z$$ Not in database $4$ 4.0.735488.6 $$\Z/4\Z$$ Not in database $8$ 8.4.16708604563456.43 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.156332410863616.15 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ 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.