# Properties

 Label 435344.bc4 Conductor $435344$ Discriminant $-1.298\times 10^{21}$ j-invariant $$-\frac{1006057824354}{131332646081}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -448019, -1737400782])

gp: E = ellinit([0, 0, 0, -448019, -1737400782])

magma: E := EllipticCurve([0, 0, 0, -448019, -1737400782]);

$$y^2=x^3-448019x-1737400782$$

## Mordell-Weil group structure

$\Z/{2}\Z$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(1326, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(1326, 0\right)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$435344$$ = $2^{4} \cdot 7 \cdot 13^{2} \cdot 23$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-1298263240903855163392$ = $-1 \cdot 2^{11} \cdot 7 \cdot 13^{14} \cdot 23$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{1006057824354}{131332646081}$$ = $-1 \cdot 2 \cdot 3^{3} \cdot 7^{-1} \cdot 11^{3} \cdot 13^{-8} \cdot 23^{-1} \cdot 241^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $2.7307109465746154773509847551\dots$ Stable Faltings height: $0.81285135233056390902511158964\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $0$ sage: E.regulator()  magma: Regulator(E); Regulator: $1$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.067839727274212311860269348962\dots$ 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: $8$  = $2\cdot1\cdot2^{2}\cdot1$ 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: $16$ = $4^2$ (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: $L(E,1)$ ≈ $2.1708712727747939795286191667924701436$

## Modular invariants

Modular form 435344.2.a.bc

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);

$$q + 2q^{5} - q^{7} - 3q^{9} + 2q^{17} + 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 12042240 $\Gamma_0(N)$-optimal: no Manin constant: 1 (conditional*)
* The Manin constant is correct provided that curve 435344.bc3 is optimal.

## 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$(j)_{-}$
$2$ $2$ $I_3^{*}$ Additive 1 4 11 0
$7$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$13$ $4$ $I_8^{*}$ Additive 1 2 14 8
$23$ $1$ $I_{1}$ 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 8.12.0.9

## $p$-adic regulators

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

All $p$-adic regulators are identically $1$ since the rank is $0$.

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 435344.bc consists of 3 curves linked by isogenies of degrees dividing 4.