# Properties

 Label 101568.do1 Conductor $101568$ Discriminant $4249452269579010048$ j-invariant $\frac{12214672127}{9}$ CM no Rank $1$ Torsion Structure $\Z/{2}\Z$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 0, -37360801, -87909060673]); // or
magma: E := EllipticCurve("101568bt2");
sage: E = EllipticCurve([0, 1, 0, -37360801, -87909060673]) # or
sage: E = EllipticCurve("101568bt2")
gp: E = ellinit([0, 1, 0, -37360801, -87909060673]) \\ or
gp: E = ellinit("101568bt2")

$y^2 = x^{3} + x^{2} - 37360801 x - 87909060673$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

 $P$ = $\left(\frac{4162585045846422341}{42406599841225}, -\frac{8475844931360907731792033964}{276153262397051642875}\right)$ $\hat{h}(P)$ ≈ 41.6655383002

## Torsion generators

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

$\left(-3527, 0\right)$

## Integral points

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

$\left(-3527, 0\right)$

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] $N$ = $101568$ = $2^{6} \cdot 3 \cdot 23^{2}$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc $\Delta$ = $4249452269579010048$ = $2^{18} \cdot 3^{2} \cdot 23^{9}$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j $j$ = $\frac{12214672127}{9}$ = $3^{-2} \cdot 7^{6} \cdot 47^{3}$ $\text{End} (E)$ = $\Z$ (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() $\text{Reg}$ ≈ $41.6655383002$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] $\Omega$ ≈ $0.0610654452751$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] $\prod_p c_p$ = $16$  = $2^{2}\cdot2\cdot2$ 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 101568.2.1.do

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)

$q + q^{3} + 4q^{5} - 4q^{7} + q^{9} + 2q^{13} + 4q^{15} - 4q^{17} + 4q^{19} + O(q^{20})$

### Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
9043968 : 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])

$L'(E,1)$ ≈ $10.1772985957$

## 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$(j)_{-}$
$2$ $4$ $I_8^{*}$ Additive 1 6 18 0
$3$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$23$ $2$ $III^{*}$ Additive -1 2 9 0

## 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 X43.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 3 \\ 2 & 3 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 5 \\ 0 & 1 \end{array}\right)$ 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
$2$ 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 add split ordinary ordinary ss ordinary ordinary ordinary add ordinary ordinary ordinary ordinary ordinary ss - 2 1 1 1,1 1 1 1 - 1 1 1 1 1 1,1 - 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 non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 101568.do consists of 2 curves linked by isogenies of degree 2.

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

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $\Q(\sqrt{23})$ $\Z/2\Z \times \Z/2\Z$ Not in database
4 4.0.778688.1 $\Z/4\Z$ 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.