Properties

Label 4730k1
Conductor 4730
Discriminant -360317791790000000
j-invariant \( -\frac{19237750463016353596082481}{360317791790000000} \)
CM no
Rank 1
Torsion Structure \(\Z/{7}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, -1, 1, -5582262, 5077966149]); // or
 
magma: E := EllipticCurve("4730k1");
 
sage: E = EllipticCurve([1, -1, 1, -5582262, 5077966149]) # or
 
sage: E = EllipticCurve("4730k1")
 
gp: E = ellinit([1, -1, 1, -5582262, 5077966149]) \\ or
 
gp: E = ellinit("4730k1")
 

\( y^2 + x y + y = x^{3} - x^{2} - 5582262 x + 5077966149 \)

Mordell-Weil group structure

\(\Z\times \Z/{7}\Z\)

Infinite order Mordell-Weil generator and height

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

\(P\) =  \( \left(-2493, 60371\right) \)
\(\hat{h}(P)\) ≈  0.982057305149

Torsion generators

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

\( \left(257, 60371\right) \)

Integral points

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

\( \left(-2493, 60371\right) \), \( \left(257, 60371\right) \), \( \left(907, 27121\right) \), \( \left(1357, -129\right) \), \( \left(1377, 171\right) \), \( \left(1467, 5921\right) \), \( \left(1643, 17603\right) \), \( \left(2237, 60371\right) \), \( \left(4129, 226867\right) \), \( \left(17197, 2226271\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 4730 \)  =  \(2 \cdot 5 \cdot 11 \cdot 43\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(-360317791790000000 \)  =  \(-1 \cdot 2^{7} \cdot 5^{7} \cdot 11^{7} \cdot 43^{2} \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( -\frac{19237750463016353596082481}{360317791790000000} \)  =  \(-1 \cdot 2^{-7} \cdot 3^{3} \cdot 5^{-7} \cdot 11^{-7} \cdot 37^{3} \cdot 43^{-2} \cdot 2413951^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(1\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(0.982057305149\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(0.278144148002\)
magma: TamagawaNumbers(E);
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
Tamagawa product: \( 686 \)  = \( 7\cdot7\cdot7\cdot2 \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(7\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 4730.2.a.d

magma: ModularForm(E);
 
sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 

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

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 219520
\( \Gamma_0(N) \)-optimal: yes
Manin constant: 1

Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 

\( L'(E,1) \) ≈ \( 3.82414889401 \)

Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(7\) \( I_{7} \) Split multiplicative -1 1 7 7
\(5\) \(7\) \( I_{7} \) Split multiplicative -1 1 7 7
\(11\) \(7\) \( I_{7} \) Split multiplicative -1 1 7 7
\(43\) \(2\) \( I_{2} \) Split multiplicative -1 1 2 2

Galois representations

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

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 
sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 

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

prime Image of Galois representation
\(7\) B.1.1

$p$-adic data

$p$-adic regulators

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

Note: \(p\)-adic regulator data only exists for primes \(p\ge5\) of good ordinary reduction.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type split ss split ordinary split ss ordinary ordinary ordinary ordinary ordinary ordinary ss split ordinary
$\lambda$-invariant(s) 3 1,1 4 7 2 1,1 1 1 3 1 1 1 1,1 2 1
$\mu$-invariant(s) 0 0,0 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 \(d=\) 7.
Its isogeny class 4730k consists of 2 curves linked by isogenies of degree 7.

Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{7}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.440.1 \(\Z/14\Z\) Not in database
6 6.0.85184000.1 \(\Z/2\Z \times \Z/14\Z\) Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.