Properties

Label 28224.es4
Conductor $28224$
Discriminant $-3.888\times 10^{20}$
j-invariant \( -\frac{4354703137}{17294403} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -960204, 1015494032])
 
gp: E = ellinit([0, 0, 0, -960204, 1015494032])
 
magma: E := EllipticCurve([0, 0, 0, -960204, 1015494032]);
 

\(y^2=x^3-960204x+1015494032\)  Toggle raw display

Mordell-Weil group structure

$\Z/{2}\Z$

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(-1316, 0\right) \)  Toggle raw display

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\( \left(-1316, 0\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 28224 \)  =  $2^{6} \cdot 3^{2} \cdot 7^{2}$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-388831342839926095872 $  =  $-1 \cdot 2^{18} \cdot 3^{7} \cdot 7^{14} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{4354703137}{17294403} \)  =  $-1 \cdot 3^{-1} \cdot 7^{-8} \cdot 23^{3} \cdot 71^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.6361873730925692331344130270\dots$
Stable Faltings height: $0.074205383390939770758265854628\dots$

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: $0.14743606245666539577854373225\dots$
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: $ 32 $  = $ 2\cdot2^{2}\cdot2^{2} $
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) $ ≈ $ 1.1794884996533231662283498580268883419 $

Modular invariants

Modular form 28224.2.a.es

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);
 

\( q + 2 q^{5} - 4 q^{11} - 2 q^{13} - 6 q^{17} - 4 q^{19} + O(q^{20}) \)  Toggle raw display

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 3 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$(j)_{-}$
$2$ $2$ $I_{8}^{*}$ Additive -1 6 18 0
$3$ $4$ $I_{1}^{*}$ Additive -1 2 7 1
$7$ $4$ $I_{8}^{*}$ Additive -1 2 14 8

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 16.48.0.124

$p$-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 $0$.

Iwasawa invariants

$p$ 2 3 7
Reduction type add add add
$\lambda$-invariant(s) - - -
$\mu$-invariant(s) - - -

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

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, 4 and 8.
Its isogeny class 28224.es consists of 4 curves linked by isogenies of degrees dividing 8.

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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-3}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{42}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{-14}) \) \(\Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-3}, \sqrt{-14})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{2}, \sqrt{21})\) \(\Z/8\Z\) Not in database
$4$ \(\Q(\sqrt{-6}, \sqrt{-7})\) \(\Z/8\Z\) Not in database
$8$ 8.0.114709561344.21 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.0.458838245376.35 \(\Z/8\Z\) Not in database
$8$ 8.0.796594176.2 \(\Z/2\Z \times \Z/8\Z\) Not in database
$8$ 8.0.7341411926016.73 \(\Z/16\Z\) Not in database
$8$ 8.2.85365421682688.21 \(\Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/4\Z \times \Z/4\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/16\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/16\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/6\Z\) 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.