Properties

Label 3675.g1
Conductor \(3675\)
Discriminant \(-297675\)
j-invariant \( -\frac{46585}{243} \)
CM no
Rank \(1\)
Torsion Structure \(\mathrm{Trivial}\)

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This elliptic curve is congruent modulo 17 to the elliptic curve 47775.be1, 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, -8, 27]); // or
magma: E := EllipticCurve("3675k1");
sage: E = EllipticCurve([1, 0, 0, -8, 27]) # or
sage: E = EllipticCurve("3675k1")
gp: E = ellinit([1, 0, 0, -8, 27]) \\ or
gp: E = ellinit("3675k1")

\( y^2 + x y = x^{3} - 8 x + 27 \)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(1, 4\right) \)
\(\hat{h}(P)\) ≈  0.201144133902

Integral points

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

\( \left(-2, 7\right) \), \( \left(1, 4\right) \), \( \left(13, 40\right) \)

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

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
\( N \)  =  \( 3675 \)  =  \(3 \cdot 5^{2} \cdot 7^{2}\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
\(\Delta\)  =  \(-297675 \)  =  \(-1 \cdot 3^{5} \cdot 5^{2} \cdot 7^{2} \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
\(j \)  =  \( -\frac{46585}{243} \)  =  \(-1 \cdot 3^{-5} \cdot 5 \cdot 7 \cdot 11^{3}\)
\( \text{End} (E) \)  =  \(\Z\)   (no Complex Multiplication)
\( \text{ST} (E) \)  =  $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
sage: E.rank()
\( r \)  =  \(1\)
magma: Regulator(E);
sage: E.regulator()
\( \text{Reg} \)  ≈  \(0.201144133902\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
\( \Omega \)  ≈  \(2.66136091092\)
magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
\( \prod_p c_p \)  =  \( 5 \)  = \( 5\cdot1\cdot1 \)
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 3675.2.1.g

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)

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

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

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
360 : 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])

\( L'(E,1) \) ≈ \( 2.67658567713 \)

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

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]

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 ordinary split add add ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary
$\lambda$-invariant(s) 1 2 - - 1,1 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

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

Isogenies

This curve has no rational isogenies. Its isogeny class 3675.g consists of this curve only.