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Properties

Label 570.b1
Conductor $570$
Discriminant $5540400$
j-invariant \( \frac{468898230633769}{5540400} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -1618, 24388])
 
gp: E = ellinit([1, 1, 0, -1618, 24388])
 
magma: E := EllipticCurve([1, 1, 0, -1618, 24388]);
 

Minimal equation

Minimal equation

Simplified equation

\(y^2+xy=x^3+x^2-1618x+24388\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz=x^3+x^2z-1618xz^2+24388z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-2097603x+1169307198\) Copy content Toggle raw display (homogenize, minimize)

Mordell-Weil group structure

\(\Z \oplus \Z/{2}\Z\)

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(16, 46\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $0.31164954389922317457019348119$

Torsion generators

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

\( \left(\frac{91}{4}, -\frac{91}{8}\right) \) Copy content Toggle raw display

Integral points

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

\( \left(-11, 208\right) \), \( \left(-11, -197\right) \), \( \left(16, 46\right) \), \( \left(16, -62\right) \), \( \left(23, -10\right) \), \( \left(23, -13\right) \), \( \left(24, -2\right) \), \( \left(24, -22\right) \), \( \left(84, 658\right) \), \( \left(84, -742\right) \) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 570 \)  =  $2 \cdot 3 \cdot 5 \cdot 19$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $5540400 $  =  $2^{4} \cdot 3^{6} \cdot 5^{2} \cdot 19 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{468898230633769}{5540400} \)  =  $2^{-4} \cdot 3^{-6} \cdot 5^{-2} \cdot 19^{-1} \cdot 77689^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.44399078785278025126495506941\dots$
Stable Faltings height: $0.44399078785278025126495506941\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.31164954389922317457019348119\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $2.1867251700822314888159175386\dots$
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\cdot2\cdot2\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) $ ≈ $ 1.3629838037781573305271615791 $

Modular invariants

Modular form   570.2.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);
 

\( q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + 2 q^{7} - q^{8} + q^{9} + q^{10} - 6 q^{11} - q^{12} - 2 q^{14} + q^{15} + q^{16} + 2 q^{17} - q^{18} - q^{19} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 384
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is semistable. There are 4 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$(j)_{-}$
$2$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$3$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6
$5$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$19$ $1$ $I_{1}$ Non-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 2.3.0.1

$p$-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 nonsplit nonsplit nonsplit ord ord ss ord nonsplit ord ord ord ord ord ord ord
$\lambda$-invariant(s) 1 1 1 1 1 1,1 1 5 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

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 570.b consists of 2 curves linked by isogenies of degree 2.

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
$2$ \(\Q(\sqrt{19}) \) \(\Z/2\Z \oplus \Z/2\Z\) Not in database
$4$ 4.0.17100.4 \(\Z/4\Z\) Not in database
$8$ 8.0.1688960160000.14 \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \oplus \Z/4\Z\) Not in database
$8$ 8.2.2850120270000.8 \(\Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \oplus \Z/6\Z\) 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.