# Properties

 Label 100010b1 Conductor 100010 Discriminant -640064000 j-invariant $$-\frac{135803877416902729}{640064000}$$ CM no Rank 1 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 1, -10709, 425632]); // or
magma: E := EllipticCurve("100010b1");
sage: E = EllipticCurve([1, 0, 1, -10709, 425632]) # or
sage: E = EllipticCurve("100010b1")
gp: E = ellinit([1, 0, 1, -10709, 425632]) \\ or
gp: E = ellinit("100010b1")

$$y^2 + x y + y = x^{3} - 10709 x + 425632$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

 $$P$$ = $$\left(\frac{235}{4}, -\frac{141}{8}\right)$$ $$\hat{h}(P)$$ ≈ 3.61170634596

## Integral points

magma: IntegralPoints(E);
sage: E.integral_points()
None

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$100010$$ = $$2 \cdot 5 \cdot 73 \cdot 137$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$-640064000$$ = $$-1 \cdot 2^{9} \cdot 5^{3} \cdot 73 \cdot 137$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$-\frac{135803877416902729}{640064000}$$ = $$-1 \cdot 2^{-9} \cdot 5^{-3} \cdot 73^{-1} \cdot 137^{-1} \cdot 514009^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E); sage: E.rank() Rank: $$1$$ magma: Regulator(E); sage: E.regulator() Regulator: $$3.61170634596$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$1.43210642197$$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$1$$  = $$1\cdot1\cdot1\cdot1$$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] Torsion order: $$1$$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 100010.2.a.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)

$$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + 2q^{7} - q^{8} - 2q^{9} + q^{10} - 4q^{11} + q^{12} + 2q^{13} - 2q^{14} - q^{15} + q^{16} + q^{17} + 2q^{18} + 3q^{19} + O(q^{20})$$

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 100224 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

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])

$$L'(E,1)$$ ≈ $$5.17234785233$$

## 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$$(j)_{-}$$
$$2$$ $$1$$ $$I_{9}$$ Non-split multiplicative 1 1 9 9
$$5$$ $$1$$ $$I_{3}$$ Non-split multiplicative 1 1 3 3
$$73$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$137$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1

## 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

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 73 137 nonsplit ordinary nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary split split 5 1 3 1 3 1 1 1 1 1 1 1 1 1 1 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

## Isogenies

This curve has no rational isogenies. Its isogeny class 100010b consists of this curve only.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.400040.1 $$\Z/2\Z$$ Not in database
6 $$x^{6}$$ $$\mathstrut +\mathstrut 146 x^{4}$$ $$\mathstrut -\mathstrut 4132 x^{3}$$ $$\mathstrut +\mathstrut 105339 x^{2}$$ $$\mathstrut -\mathstrut 901696 x$$ $$\mathstrut +\mathstrut 5168446$$ $$\Z/2\Z \times \Z/2\Z$$ 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.