Properties

Label 144026.d1
Conductor \(144026\)
Discriminant \(205381076\)
j-invariant \( \frac{1129315234499353}{205381076} \)
CM no
Rank \(1\)
Torsion Structure \(\Z/{2}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 0, -2169, 37985]); // or
magma: E := EllipticCurve("144026k2");
sage: E = EllipticCurve([1, 1, 0, -2169, 37985]) # or
sage: E = EllipticCurve("144026k2")
gp: E = ellinit([1, 1, 0, -2169, 37985]) \\ or
gp: E = ellinit("144026k2")

\( y^2 + x y = x^{3} + x^{2} - 2169 x + 37985 \)

Mordell-Weil group structure

\(\Z\times \Z/{2}\Z\)

Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(-43, 254\right) \)
\(\hat{h}(P)\) ≈  2.56345176351

Torsion generators

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

\( \left(\frac{107}{4}, -\frac{107}{8}\right) \)

Integral points

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

\( \left(-43, 254\right) \)

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

Invariants

magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
\( N \)  =  \( 144026 \)  =  \(2 \cdot 23 \cdot 31 \cdot 101\)
magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
\(\Delta\)  =  \(205381076 \)  =  \(2^{2} \cdot 23^{2} \cdot 31^{2} \cdot 101 \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
\(j \)  =  \( \frac{1129315234499353}{205381076} \)  =  \(2^{-2} \cdot 11^{3} \cdot 23^{-2} \cdot 31^{-2} \cdot 101^{-1} \cdot 9467^{3}\)
\( \text{End} (E) \)  =  \(\Z\)   (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()
\( \text{Reg} \)  ≈  \(2.56345176351\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
\( \Omega \)  ≈  \(1.72791224963\)
magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
\( \prod_p c_p \)  =  \( 8 \)  = \( 2\cdot2\cdot2\cdot1 \)
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
\( \#E_{\text{tor}} \)  = \(2\)
magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
Ш\(_{\text{an}} \)  =   \(1\) (exact)

Modular invariants

Modular form 144026.2.1.d

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)

\( q - q^{2} + 2q^{3} + q^{4} + 2q^{5} - 2q^{6} - 2q^{7} - q^{8} + q^{9} - 2q^{10} + 2q^{12} - 2q^{13} + 2q^{14} + 4q^{15} + q^{16} + 6q^{17} - q^{18} + 4q^{19} + O(q^{20}) \)

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

Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
111616 : curve is not \( \Gamma_0(N) \)-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])

\( L'(E,1) \) ≈ \( 8.858839407 \)

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\((j)_{-}\)
\(2\) \(2\) \( I_{2} \) Non-split multiplicative 1 1 2 2
\(23\) \(2\) \( I_{2} \) Split multiplicative -1 1 2 2
\(31\) \(2\) \( I_{2} \) Split multiplicative -1 1 2 2
\(101\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1

Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X6.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right)$ and has index 3.

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
\(2\) B

$p$-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.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 101
Reduction type nonsplit ordinary ordinary ordinary ss ordinary ordinary ordinary split ss split ordinary ordinary ordinary ordinary nonsplit
$\lambda$-invariant(s) 9 1 5 3 1,1 1 3 1 2 1,1 2 1 1 1 1 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 \(d=\) 2.
Its isogeny class 144026.d consists of 2 curves linked by isogenies of degree 2.