Properties

Label 455175dj1
Conductor 455175
Discriminant -617706806840812398727781982421875
j-invariant \( -\frac{11926249134908509075308544}{2246680441062421875} \)
CM no
Rank 1
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 1, -309509332950, -66287159967293969]) # or
 
sage: E = EllipticCurve("455175dj1")
 
gp: E = ellinit([0, 0, 1, -309509332950, -66287159967293969]) \\ or
 
gp: E = ellinit("455175dj1")
 
magma: E := EllipticCurve([0, 0, 1, -309509332950, -66287159967293969]); // or
 
magma: E := EllipticCurve("455175dj1");
 

\( y^2 + y = x^{3} - 309509332950 x - 66287159967293969 \)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(\frac{5787501113364489761044908265}{367674268745071073536}, \frac{440008954605836444437205706250198369251467}{7050093163382738951992787849216}\right) \)
\(\hat{h}(P)\) ≈  53.56564516850286

Integral points

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

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 455175 \)  =  \(3^{2} \cdot 5^{2} \cdot 7 \cdot 17^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-617706806840812398727781982421875 \)  =  \(-1 \cdot 3^{16} \cdot 5^{13} \cdot 7^{3} \cdot 17^{11} \)
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: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(53.5656451685029\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.00320034069158087\)
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: \( 48 \)  = \( 2\cdot2\cdot3\cdot2^{2} \)
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 455175.2.a.dj

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);
 

\( q - 2q^{4} + q^{7} + 2q^{11} + 5q^{13} + 4q^{16} + 2q^{19} + O(q^{20}) \)

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 2322432000
\( \Gamma_0(N) \)-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);
 

\( L'(E,1) \) ≈ \( 8.228559067370007 \)

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\((j)_{-}\)
\(3\) \(2\) \( I_10^{*} \) Additive -1 2 16 10
\(5\) \(2\) \( I_7^{*} \) Additive 1 2 13 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]
 

\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has no rational isogenies. Its isogeny class 455175dj consists of this curve only.