# Properties

 Label 229320dr2 Conductor $229320$ Discriminant $1.025\times 10^{16}$ j-invariant $$\frac{11151682683009628}{40040325}$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -2955603, 1955761598])

gp: E = ellinit([0, 0, 0, -2955603, 1955761598])

magma: E := EllipticCurve([0, 0, 0, -2955603, 1955761598]);

$$y^2=x^3-2955603x+1955761598$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(959, 1820\right)$$ $$\left(1414, 24570\right)$$ $\hat{h}(P)$ ≈ $0.75567547873289744327829595472$ $2.0549862545820184357613005412$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(994, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(-1841,\pm 34020)$$, $$(127,\pm 39780)$$, $$(751,\pm 12636)$$, $$(959,\pm 1820)$$, $$(991,\pm 36)$$, $$\left(994, 0\right)$$, $$(998,\pm 286)$$, $$(1414,\pm 24570)$$, $$(6178,\pm 468504)$$, $$(14918,\pm 1810474)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$229320$$ = $2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $10252250260761600$ = $2^{10} \cdot 3^{12} \cdot 5^{2} \cdot 7^{3} \cdot 13^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{11151682683009628}{40040325}$$ = $2^{2} \cdot 3^{-6} \cdot 5^{-2} \cdot 13^{-3} \cdot 17^{6} \cdot 487^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.2898706940730765561242004381\dots$ Stable Faltings height: $0.67646436200857229296921286590\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $2$ sage: E.regulator()  magma: Regulator(E); Regulator: $1.5098460418638761747343414527\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.35634072096767738865898299451\dots$ 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: $96$  = $2\cdot2^{2}\cdot2\cdot2\cdot3$ 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: $1$ (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: $L^{(2)}(E,1)/2!$ ≈ $12.912471050591223675926838149300122142$

## Modular invariants

Modular form 229320.2.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);

$$q - q^{5} - 4q^{11} + q^{13} + 2q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 3833856 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $2$ $III^{*}$ Additive 1 3 10 0
$3$ $4$ $I_6^{*}$ Additive -1 2 12 6
$5$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$7$ $2$ $III$ Additive -1 2 3 0
$13$ $3$ $I_{3}$ Split multiplicative -1 1 3 3

## 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 $\GL(2,\Z_\ell)$ 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 229320dr 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 $K$ $2$ $$\Q(\sqrt{91})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $4$ 4.4.4013100.1 $$\Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.8.43547843233440000.2 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ 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.