# Properties

 Label 212160.cx4 Conductor $212160$ Discriminant $-8.005\times 10^{14}$ j-invariant $$\frac{931329171502}{6107473125}$$ CM no Rank $2$ Torsion structure $$\Z/{4}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, 10335, -1303263])

gp: E = ellinit([0, -1, 0, 10335, -1303263])

magma: E := EllipticCurve([0, -1, 0, 10335, -1303263]);

$$y^2=x^3-x^2+10335x-1303263$$

## Mordell-Weil group structure

$\Z^2 \times \Z/{4}\Z$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(89, 560\right)$$ $$\left(368, 7225\right)$$ $\hat{h}(P)$ ≈ $2.0714314402447355143702497483$ $3.2572983410380945617347989283$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(249, 4080\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(79, 0\right)$$, $$(89,\pm 560)$$, $$(147,\pm 1836)$$, $$(179,\pm 2500)$$, $$(249,\pm 4080)$$, $$(368,\pm 7225)$$, $$(504,\pm 11475)$$, $$(889,\pm 26640)$$, $$(1167,\pm 39984)$$, $$(2969,\pm 161840)$$, $$(8409,\pm 771120)$$

## 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: $-800518717440000$ = $-1 \cdot 2^{17} \cdot 3^{2} \cdot 5^{4} \cdot 13 \cdot 17^{4}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{931329171502}{6107473125}$$ = $2 \cdot 3^{-2} \cdot 5^{-4} \cdot 13^{-1} \cdot 17^{-4} \cdot 23^{3} \cdot 337^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $1.5414397647469985391735141105\dots$ Stable Faltings height: $0.55948125895374268416576860510\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $2$ sage: E.regulator()  magma: Regulator(E); Regulator: $6.1250857100171428135927420140\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.25104992467891980438224483299\dots$ 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: $128$  = $2^{2}\cdot2\cdot2^{2}\cdot1\cdot2^{2}$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $4$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (rounded) 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: $L^{(2)}(E,1)/2!$ ≈ $12.301618449213853872951406795655129736$

## Modular invariants

Modular form 212160.2.a.cx

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);

$$q - q^{3} + q^{5} + q^{9} - 4 q^{11} + q^{13} - q^{15} + q^{17} - 8 q^{19} + O(q^{20})$$

For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 786432 $\Gamma_0(N)$-optimal: no 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$(j)_{-}$
$2$ $4$ $I_{7}^{*}$ Additive 1 6 17 0
$3$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$5$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$13$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$17$ $4$ $I_{4}$ Split multiplicative -1 1 4 4

## 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 4.12.0.7

## $p$-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 212160.cx 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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-26})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ 4.2.1731225600.30 $$\Z/8\Z$$ Not in database $8$ 8.0.6559413791883264.35 $$\Z/4\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/8\Z$$ Not in database $8$ Deg 8 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/12\Z$$ 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.