# Properties

 Label 448b1 Conductor $448$ Discriminant $-114688$ j-invariant $$\frac{432}{7}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Simplified equation

 $$y^2=x^3+4x-16$$ y^2=x^3+4x-16 (homogenize, simplify) $$y^2z=x^3+4xz^2-16z^3$$ y^2z=x^3+4xz^2-16z^3 (dehomogenize, simplify) $$y^2=x^3+4x-16$$ y^2=x^3+4x-16 (homogenize, minimize)

sage: E = EllipticCurve([0, 0, 0, 4, -16])

gp: E = ellinit([0, 0, 0, 4, -16])

magma: E := EllipticCurve([0, 0, 0, 4, -16]);

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);

 $P$ = $$\left(4, 8\right)$$ (4, 8) $\hat{h}(P)$ ≈ $1.0691085767972255605925653628$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(2, 0\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(2, 0\right)$$, $$(4,\pm 8)$$, $$(10,\pm 32)$$

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$448$$ = $2^{6} \cdot 7$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $-114688$ = $-1 \cdot 2^{14} \cdot 7$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{432}{7}$$ = $2^{4} \cdot 3^{3} \cdot 7^{-1}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.34860641707856014305515959772\dots$ Stable Faltings height: $-1.1572781277318296707085970728\dots$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $1.0691085767972255605925653628\dots$ sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Real period: $1.6236666926210273234649262972\dots$ 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: $4$  = $2^{2}\cdot1$ 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: $1$ (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: $L'(E,1)$ ≈ $1.7358759869411248186869537581$

## Modular invariants

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);

$$q - 2 q^{5} + q^{7} - 3 q^{9} - 4 q^{11} - 2 q^{13} - 6 q^{17} + 8 q^{19} + O(q^{20})$$

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 32 $\Gamma_0(N)$-optimal: yes Manin constant: 1

## Local data

This elliptic curve is not semistable. There are 2 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$(j)_{-}$
$2$ $4$ $I_{4}^{*}$ Additive -1 6 14 0
$7$ $1$ $I_{1}$ 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 8.24.0.101

The image of the adelic Galois representation has level $56$, index $48$, and genus $0$.

## $p$-adic regulators

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

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

## Iwasawa invariants

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 add ss ord split ord ord ord ord ss ord ord ord ord ord ord - 3,1 1 2 1 3 1 1 1,1 1 1 1 1 1 1 - 0,0 0 0 0 0 0 0 0,0 0 0 0 0 0 0

An entry - indicates that the invariants are not computed because the reduction is additive.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 448b 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-7})$$ $$\Z/2\Z \oplus \Z/2\Z$$ 2.0.7.1-28672.7-m2 $2$ $$\Q(\sqrt{14})$$ $$\Z/4\Z$$ 2.2.56.1-56.1-d2 $2$ $$\Q(\sqrt{-2})$$ $$\Z/4\Z$$ 2.0.8.1-392.1-a1 $4$ $$\Q(\sqrt{-2}, \sqrt{-7})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ 4.2.87808.2 $$\Z/8\Z$$ Not in database $8$ 8.0.1927561216.7 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.3211264.1 $$\Z/8\Z$$ Not in database $8$ 8.0.7710244864.1 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.2.344128684032.22 $$\Z/6\Z$$ Not in database $16$ 16.0.59447875862838378496.4 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $16$ 16.0.24759631762948096.2 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/16\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/12\Z$$ Not in database $16$ Deg 16 $$\Z/12\Z$$ 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.