# Properties

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

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## 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)

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

### 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

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 101 nonsplit ordinary ordinary ordinary ss ordinary ordinary ordinary split ss split ordinary ordinary ordinary ordinary nonsplit 9 1 5 3 1,1 1 3 1 2 1,1 2 1 1 1 1 1 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.