Properties

Label 485520.dt1
Conductor 485520
Discriminant -222123639468419650825920000000
j-invariant \( -\frac{11926249134908509075308544}{2246680441062421875} \)
CM no
Rank 0
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -22009552565, -1256993622862275]) # or
 
sage: E = EllipticCurve("485520dt1")
 
gp: E = ellinit([0, -1, 0, -22009552565, -1256993622862275]) \\ or
 
gp: E = ellinit("485520dt1")
 
magma: E := EllipticCurve([0, -1, 0, -22009552565, -1256993622862275]); // or
 
magma: E := EllipticCurve("485520dt1");
 

\( y^2 = x^{3} - x^{2} - 22009552565 x - 1256993622862275 \)

Mordell-Weil group structure

Trivial

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 
None

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 485520 \)  =  \(2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 17^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-222123639468419650825920000000 \)  =  \(-1 \cdot 2^{12} \cdot 3^{10} \cdot 5^{7} \cdot 7^{3} \cdot 17^{11} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{11926249134908509075308544}{2246680441062421875} \)  =  \(-1 \cdot 2^{15} \cdot 3^{-10} \cdot 5^{-7} \cdot 7^{-3} \cdot 17^{-5} \cdot 7139777^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Rank: \(0\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(1\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.00619743310034131\)
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: \( 168 \)  = \( 1\cdot2\cdot7\cdot3\cdot2^{2} \)
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)

Modular invariants

Modular form 485520.2.a.dt

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);
 

\( q - q^{3} + q^{5} + q^{7} + q^{9} + 2q^{11} - 5q^{13} - q^{15} - 2q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 870912000
\( \Gamma_0(N) \)-optimal: yes
Manin constant: 1

Special L-value

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);
 

\( L(E,1) \) ≈ \( 1.0411687608573406 \)

Local data

This elliptic curve is not semistable.

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\((j)_{-}\)
\(2\) \(1\) \( II^{*} \) Additive -1 4 12 0
\(3\) \(2\) \( I_{10} \) Non-split multiplicative 1 1 10 10
\(5\) \(7\) \( I_{7} \) Split multiplicative -1 1 7 7
\(7\) \(3\) \( I_{3} \) Split multiplicative -1 1 3 3
\(17\) \(4\) \( I_5^{*} \) Additive 1 2 11 5

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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 mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) .

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]
 

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has no rational isogenies. Its isogeny class 485520.dt consists of this curve only.