Properties

Label 47775.be1
Conductor \(47775\)
Discriminant \(-95370835511179250711325\)
j-invariant \( \frac{48412981936758748562855}{77853743274432041397} \)
CM no
Rank \(0\)
Torsion Structure \(\mathrm{Trivial}\)

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This elliptic curve is congruent modulo 17 to 3675.g1, meaning that the 17-torsion subgroups of these elliptic curves are isomorphic as Galois modules. The isomorphism is anti-symplectic; there are no known examples of symplectic mod 17 congruences. All known examples of 17-congruences correspond to quadratic twists of this minimal pair, and there are no known examples of congruences modulo primes greater than 17.

See N. Billerey, "On some remarkable congruences between two eliptic curves" [arXiv:1605.09205] and N. Freitas and A. Kraus, "On symplectic isomorphisms of the $p$-torsion of elliptic curves", [arXiv:1607.01218] .

Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, 8124402, -11887136703]); // or
magma: E := EllipticCurve("47775cq1");
sage: E = EllipticCurve([1, 0, 0, 8124402, -11887136703]) # or
sage: E = EllipticCurve("47775cq1")
gp: E = ellinit([1, 0, 0, 8124402, -11887136703]) \\ or
gp: E = ellinit("47775cq1")

\( y^2 + x y = x^{3} + 8124402 x - 11887136703 \)

Mordell-Weil group structure

Trivial

Integral points

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

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
\( N \)  =  \( 47775 \)  =  \(3 \cdot 5^{2} \cdot 7^{2} \cdot 13\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
\(\Delta\)  =  \(-95370835511179250711325 \)  =  \(-1 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13^{17} \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
\(j \)  =  \( \frac{48412981936758748562855}{77853743274432041397} \)  =  \(3^{-2} \cdot 5 \cdot 7 \cdot 13^{-17} \cdot 19^{3} \cdot 41^{3} \cdot 14303^{3}\)
\( \text{End} (E) \)  =  \(\Z\)   (no Complex Multiplication)
\( \text{ST} (E) \)  =  $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
sage: E.rank()
\( r \)  =  \(0\)
magma: Regulator(E);
sage: E.regulator()
\( \text{Reg} \)  =  \(1\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
\( \Omega \)  ≈  \(0.0563231044046\)
magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
\( \prod_p c_p \)  =  \( 34 \)  = \( 2\cdot1\cdot1\cdot17 \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
\( \#E_{\text{tor}} \)  = \(1\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Ш\(_{\text{an}} \)  =   \(1\) (exact)

Modular invariants

Modular form 47775.2.1.be

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

\( q - q^{2} + q^{3} - q^{4} - q^{6} + 3q^{8} + q^{9} - q^{12} + q^{13} - q^{16} + 2q^{17} - q^{18} - q^{19} + O(q^{20}) \)

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

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
3818880 : curve is \( \Gamma_0(N) \)-optimal

Special L-value attached to the curve

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])

\( L(E,1) \) ≈ \( 1.91498554976 \)

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\((j)_{-}\)
\(3\) \(2\) \( I_{2} \) Split multiplicative -1 1 2 2
\(5\) \(1\) \( II \) Additive 1 2 2 0
\(7\) \(1\) \( II \) Additive -1 2 2 0
\(13\) \(17\) \( I_{17} \) Split multiplicative -1 1 17 17

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 \( p \) .

$p$-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 \(0\).

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ordinary split add add ss split ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary
$\lambda$-invariant(s) 2 1 - - 0,0 1 2 0 0 0 0 0 0,0 0 0
$\mu$-invariant(s) 0 0 - - 0,0 0 0 0 0 0 0 0 0,0 0 0

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has no rational isogenies. Its isogeny class 47775.be consists of this curve only.