# Properties

 Label 129.b2 Conductor $129$ Discriminant $1347921$ j-invariant $\frac{2845178713}{1347921}$ CM no Rank $0$ Torsion Structure $\Z/{2}\Z \times \Z/{2}\Z$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 1, -30, -29]); // or
magma: E := EllipticCurve("129b1");
sage: E = EllipticCurve([1, 0, 1, -30, -29]) # or
sage: E = EllipticCurve("129b1")
gp: E = ellinit([1, 0, 1, -30, -29]) \\ or
gp: E = ellinit("129b1")

$y^2 + x y + y = x^{3} - 30 x - 29$

## Mordell-Weil group structure

$\Z/{2}\Z \times \Z/{2}\Z$

## Torsion generators

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

$\left(-1, 0\right)$, $\left(\frac{23}{4}, -\frac{27}{8}\right)$

## Integral points

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

$\left(-5, 2\right)$, $\left(-1, 0\right)$

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] $N$ = $129$ = $3 \cdot 43$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc $\Delta$ = $1347921$ = $3^{6} \cdot 43^{2}$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j $j$ = $\frac{2845178713}{1347921}$ = $3^{-6} \cdot 13^{3} \cdot 43^{-2} \cdot 109^{3}$ $\text{End} (E)$ = $\Z$ (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] $\Omega$ ≈ $2.14552861305$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] $\prod_p c_p$ = $12$  = $( 2 \cdot 3 )\cdot2$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] $\#E_{\text{tor}}$ = $4$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Ш$_{\text{an}}$ = $1$ (exact)

## Modular invariants

### Modular form129.2.1.b

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)

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

### Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
15 : 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])

$L(E,1)$ ≈ $1.60914645979$

## 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$(j)_{-}$
$3$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$43$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

## 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 X8d.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^2\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 2 \\ 0 & 1 \end{array}\right)$ and has index 12.

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
$2$ Cs

## $p$-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

$p$ Reduction type $\lambda$-invariant(s) 2 3 43 ordinary split nonsplit 0 1 0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 129.b consists of 4 curves linked by isogenies of degrees dividing 4.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $\Q(\sqrt{-3})$ $\Z/2\Z \times \Z/4\Z$ 2.0.3.1-5547.2-b2
4 $\Q(\sqrt{3}, \sqrt{43})$ $\Z/2\Z \times \Z/4\Z$ Not in database
$\Q(i, \sqrt{43})$ $\Z/2\Z \times \Z/4\Z$ 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.