Properties

Label 28224.co1
Conductor $28224$
Discriminant $-2.621\times 10^{15}$
j-invariant \( -\frac{6329617441}{279936} \)
CM no
Rank $1$
Torsion structure trivial

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -81228, -9244816])
 
gp: E = ellinit([0, 0, 0, -81228, -9244816])
 
magma: E := EllipticCurve([0, 0, 0, -81228, -9244816]);
 

\(y^2=x^3-81228x-9244816\)  Toggle raw display

Mordell-Weil group structure

$\Z$

Infinite order Mordell-Weil generator and height

sage: E.gens()
 
magma: Generators(E);
 

$P$ =  \(\left(11470, 1228032\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $4.9707619045705285354198196661$

Integral points

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

\((11470,\pm 1228032)\)  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: $-2621333531787264 $  =  $-1 \cdot 2^{25} \cdot 3^{13} \cdot 7^{2} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{6329617441}{279936} \)  =  $-1 \cdot 2^{-7} \cdot 3^{-7} \cdot 7 \cdot 967^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.7235034786606267806861845069\dots$
Stable Faltings height: $-0.18984179468923157998817841766\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $4.9707619045705285354198196661\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.14103329488756330322147125155\dots$
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^{2}\cdot2\cdot1 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $1$
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) $ ≈ $ 5.6083434360252892017545824313752505433 $

Modular invariants

Modular form 28224.2.a.co

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);
 

\( q - q^{5} + 5q^{11} - 4q^{17} + 8q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 129024
$ \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$(j)_{-}$
$2$ $4$ $I_{15}^{*}$ Additive 1 6 25 7
$3$ $2$ $I_7^{*}$ Additive -1 2 13 7
$7$ $1$ $II$ Additive -1 2 2 0

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
$7$ 7B.6.1 7.24.0.1

$p$-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.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type add add ordinary add ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary
$\lambda$-invariant(s) - - 1 - 1 1,1 1 1 1 1 1 1 1,1 1 1
$\mu$-invariant(s) - - 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 $d=$ 7.
Its isogeny class 28224.co consists of 2 curves linked by isogenies of degree 7.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-6}) \) \(\Z/7\Z\) Not in database
$3$ 3.1.1176.1 \(\Z/2\Z\) Not in database
$6$ 6.0.33191424.2 \(\Z/2\Z \times \Z/14\Z\) Not in database
$8$ 8.2.341461686730752.5 \(\Z/3\Z\) Not in database
$12$ Deg 12 \(\Z/4\Z\) Not in database
$16$ Deg 16 \(\Z/21\Z\) 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.