Properties

Label 63870.o2
Conductor 63870
Discriminant 97645976616960000000
j-invariant \( \frac{5105817686570071165887579841}{97645976616960000000} \)
CM no
Rank 1
Torsion Structure \(\Z/{7}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -35874060, 82698195600]); // or
magma: E := EllipticCurve("63870p1");
sage: E = EllipticCurve([1, 0, 0, -35874060, 82698195600]) # or
sage: E = EllipticCurve("63870p1")
gp: E = ellinit([1, 0, 0, -35874060, 82698195600]) \\ or
gp: E = ellinit("63870p1")

\( y^2 + x y = x^{3} - 35874060 x + 82698195600 \)

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(3470, -1360\right) \)
\(\hat{h}(P)\) ≈  2.76903151847

Torsion generators

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

\( \left(3480, 180\right) \)

Integral points

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

\( \left(3470, -1360\right) \), \( \left(3480, 180\right) \), \( \left(3720, 25140\right) \), \( \left(4152, 70932\right) \), \( \left(4800, 142860\right) \), \( \left(7320, 457140\right) \), \( \left(16920, 2069940\right) \), \( \left(45720, 9673140\right) \)

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

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
Conductor: \( 63870 \)  =  \(2 \cdot 3 \cdot 5 \cdot 2129\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
Discriminant: \(97645976616960000000 \)  =  \(2^{28} \cdot 3^{7} \cdot 5^{7} \cdot 2129 \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( \frac{5105817686570071165887579841}{97645976616960000000} \)  =  \(2^{-28} \cdot 3^{-7} \cdot 5^{-7} \cdot 241^{3} \cdot 2129^{-1} \cdot 7145041^{3}\)
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: \(2.76903151847\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
Real period: \(0.174489095328\)
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: \( 1372 \)  = \( ( 2^{2} \cdot 7 )\cdot7\cdot7\cdot1 \)
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 63870.2.a.o

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)

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

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

magma: ModularDegree(E);
sage: E.modular_degree()
Modular degree: 4346496
\( \Gamma_0(N) \)-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])

\( L'(E,1) \) ≈ \( 13.5286425286 \)

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\((j)_{-}\)
\(2\) \(28\) \( I_{28} \) Split multiplicative -1 1 28 28
\(3\) \(7\) \( I_{7} \) Split multiplicative -1 1 7 7
\(5\) \(7\) \( I_{7} \) Split multiplicative -1 1 7 7
\(2129\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1

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 2129
Reduction type split split split ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary split
$\lambda$-invariant(s) 2 2 2 3 1 1,1 1 1 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 ?

An entry ? indicates that the invariants have not yet been computed.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 7.
Its isogeny class 63870.o 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 3.3.127740.1 \(\Z/14\Z\) Not in database
6 \(x^{6} \) \(\mathstrut -\mathstrut 2 x^{5} \) \(\mathstrut -\mathstrut 393 x^{4} \) \(\mathstrut +\mathstrut 3274 x^{3} \) \(\mathstrut +\mathstrut 3994 x^{2} \) \(\mathstrut -\mathstrut 56400 x \) \(\mathstrut +\mathstrut 29760 \) \(\Z/2\Z \times \Z/14\Z\) 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.