Properties

Label 4928.o1
Conductor $4928$
Discriminant $609266696192$
j-invariant \( \frac{15226621995131793}{2324168} \)
CM no
Rank $1$
Torsion structure \(\Z/{4}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3-330476x+73123664\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-330476xz^2+73123664z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-330476x+73123664\) Copy content Toggle raw display (homogenize, minimize)

sage: E = EllipticCurve([0, 0, 0, -330476, 73123664])
 
gp: E = ellinit([0, 0, 0, -330476, 73123664])
 
magma: E := EllipticCurve([0, 0, 0, -330476, 73123664]);
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 

Mordell-Weil group structure

\(\Z \oplus \Z/{4}\Z\)

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(325, 217\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $2.2587850369382622433455203833$

Torsion generators

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

\( \left(346, 448\right) \) Copy content Toggle raw display

Integral points

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

\((-550,\pm 9408)\), \((304,\pm 868)\), \((325,\pm 217)\), \( \left(332, 0\right) \), \((346,\pm 448)\) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 4928 \)  =  $2^{6} \cdot 7 \cdot 11$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $609266696192 $  =  $2^{21} \cdot 7^{4} \cdot 11^{2} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{15226621995131793}{2324168} \)  =  $2^{-3} \cdot 3^{3} \cdot 7^{-4} \cdot 11^{-2} \cdot 82619^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.6668938490265475202360714732\dots$
Stable Faltings height: $0.62717307818662955611022329101\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $2.2587850369382622433455203833\dots$
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: $0.71630801875515166908408806144\dots$
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^{2}\cdot2^{2}\cdot2 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $4$
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) $ ≈ $ 3.2359716692060574132823112659 $

Modular invariants

Modular form   4928.2.a.o

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);
 

\( q - 2 q^{5} + q^{7} - 3 q^{9} - q^{11} - 2 q^{13} + 2 q^{17} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 18432
$ \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$(j)_{-}$
$2$ $4$ $I_{11}^{*}$ Additive -1 6 21 3
$7$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$11$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

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 8.24.0.49

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]
 

$p$-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 ss ord split nonsplit ord ord ss ord ord ord ord ord ord ord
$\lambda$-invariant(s) - 1,3 1 2 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 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 4928.o 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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{2}) \) \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$4$ 4.4.100352.2 \(\Z/8\Z\) Not in database
$8$ 8.0.245635219456.5 \(\Z/4\Z \oplus \Z/4\Z\) Not in database
$8$ 8.0.143986855936.5 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$8$ 8.8.40282095616.2 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$8$ Deg 8 \(\Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/16\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \oplus \Z/12\Z\) 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.