Properties

Label 576.b2
Conductor $576$
Discriminant $143327232$
j-invariant \( \frac{28756228}{3} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3-2316x+42896\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3-2316xz^2+42896z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-2316x+42896\) Copy content Toggle raw display (homogenize, minimize)

sage: E = EllipticCurve([0, 0, 0, -2316, 42896])
 
gp: E = ellinit([0, 0, 0, -2316, 42896])
 
magma: E := EllipticCurve([0, 0, 0, -2316, 42896]);
 
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);
 

$P$ =  \(\left(26, 16\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $0.53963693233858955721845168329$

Torsion generators

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

\( \left(28, 0\right) \) Copy content Toggle raw display

Integral points

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

\((10,\pm 144)\), \((26,\pm 16)\), \( \left(28, 0\right) \), \((29,\pm 11)\), \((64,\pm 396)\) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 576 \)  =  $2^{6} \cdot 3^{2}$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $143327232 $  =  $2^{16} \cdot 3^{7} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{28756228}{3} \)  =  $2^{2} \cdot 3^{-1} \cdot 193^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.59710103881414288245584406309\dots$
Stable Faltings height: $-0.87640134626650570913142138398\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.53963693233858955721845168329\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $1.7607876529005999591360601728\dots$
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 $
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) $ ≈ $ 1.9003720950218899493176796283 $

Modular invariants

Modular form   576.2.a.b

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 256
$ \Gamma_0(N) $-optimal: no
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$(j)_{-}$
$2$ $4$ $I_{6}^{*}$ Additive -1 6 16 0
$3$ $2$ $I_{1}^{*}$ Additive -1 2 7 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 16.96.0.224

$p$-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 ord ss ord ord ord ord ord ord ord ord ord ord ss
$\lambda$-invariant(s) - - 1 1,1 1 1 1 1 1 3 1 1 1 1 1,1
$\mu$-invariant(s) - - 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, 4 and 8.
Its isogeny class 576.b consists of 6 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
$2$ \(\Q(\sqrt{3}) \) \(\Z/2\Z \oplus \Z/2\Z\) 2.2.12.1-768.1-l6
$2$ \(\Q(\sqrt{6}) \) \(\Z/8\Z\) 2.2.24.1-24.1-b5
$2$ \(\Q(\sqrt{2}) \) \(\Z/4\Z\) 2.2.8.1-1296.1-b6
$4$ \(\Q(\sqrt{2}, \sqrt{3})\) \(\Z/2\Z \oplus \Z/8\Z\) 4.4.2304.1-72.1-b6
$8$ 8.0.191102976.4 \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ 8.0.47775744.4 \(\Z/8\Z\) Not in database
$8$ 8.8.12230590464.1 \(\Z/2\Z \oplus \Z/16\Z\) Not in database
$8$ 8.2.11609505792.11 \(\Z/6\Z\) Not in database
$16$ 16.0.36520347436056576.1 \(\Z/4\Z \oplus \Z/8\Z\) Not in database
$16$ 16.0.336571521970697404416.2 \(\Z/2\Z \oplus \Z/8\Z\) Not in database
$16$ 16.0.1846757322198614016.2 \(\Z/2\Z \oplus \Z/16\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/24\Z\) 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.