# Properties

 Label 481338.t1 Conductor $481338$ Discriminant $3.933\times 10^{20}$ j-invariant $$\frac{3047678972871625}{304559880768}$$ 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, 0, -3289347, -2087747051])

gp: E = ellinit([1, -1, 0, -3289347, -2087747051])

magma: E := EllipticCurve([1, -1, 0, -3289347, -2087747051]);

$$y^2+xy=x^3-x^2-3289347x-2087747051$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(-1290, 3397\right)$$ $\hat{h}(P)$ ≈ $6.1440212863914431971317372928$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(-778, 389\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-1290, 3397\right)$$, $$\left(-1290, -2107\right)$$, $$\left(-778, 389\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$481338$$ = $2 \cdot 3^{2} \cdot 11^{2} \cdot 13 \cdot 17$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $393329330654331120192$ = $2^{6} \cdot 3^{10} \cdot 11^{8} \cdot 13^{4} \cdot 17$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{3047678972871625}{304559880768}$$ = $2^{-6} \cdot 3^{-4} \cdot 5^{3} \cdot 11^{-2} \cdot 13^{-4} \cdot 17^{-1} \cdot 107^{3} \cdot 271^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.6881343174553462278573884722\dots$ Stable Faltings height: $0.93988053672210611012879406476\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $6.1440212863914431971317372928\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.11282582461840754987593419222\dots$ 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: $16$  = $2\cdot2\cdot2\cdot2\cdot1$ 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: $L'(E,1)$ ≈ $2.7728170724406548614740018447421794940$

## Modular invariants

Modular form 481338.2.a.t

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);

$$q - q^{2} + q^{4} - 2q^{7} - q^{8} - q^{13} + 2q^{14} + q^{16} - q^{17} - 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 20643840 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6
$3$ $2$ $I_4^{*}$ Additive -1 2 10 4
$11$ $2$ $I_2^{*}$ Additive -1 2 8 2
$13$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$17$ $1$ $I_{1}$ 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 8.6.0.4

## $p$-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 481338.t consists of 2 curves linked by isogenies of degree 2.