# Properties

 Label 100014b1 Conductor 100014 Discriminant 1600224 j-invariant $$\frac{135559106353}{1600224}$$ CM no Rank 2 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 1, -107, 377]); // or
magma: E := EllipticCurve("100014b1");
sage: E = EllipticCurve([1, 1, 1, -107, 377]) # or
sage: E = EllipticCurve("100014b1")
gp: E = ellinit([1, 1, 1, -107, 377]) \\ or
gp: E = ellinit("100014b1")

$$y^2 + x y + y = x^{3} + x^{2} - 107 x + 377$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);
sage: E.gens()

 $$P$$ = $$\left(7, -10\right)$$ $$\left(-\frac{13}{4}, \frac{221}{8}\right)$$ $$\hat{h}(P)$$ ≈ 0.899655480558 3.84158142033

## Integral points

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

$$\left(-7, 32\right)$$, $$\left(-1, 22\right)$$, $$\left(5, -2\right)$$, $$\left(7, 2\right)$$, $$\left(79, 662\right)$$

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] Conductor: $$100014$$ = $$2 \cdot 3 \cdot 79 \cdot 211$$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc Discriminant: $$1600224$$ = $$2^{5} \cdot 3 \cdot 79 \cdot 211$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{135559106353}{1600224}$$ = $$2^{-5} \cdot 3^{-1} \cdot 11^{3} \cdot 79^{-1} \cdot 211^{-1} \cdot 467^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E); sage: E.rank() Rank: $$2$$ magma: Regulator(E); sage: E.regulator() Regulator: $$0.771377877645$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$2.6804363255$$ 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: $$5$$  = $$5\cdot1\cdot1\cdot1$$ 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: $$1$$ (rounded)

## Modular invariants

#### Modular form 100014.2.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)

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

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 25760 $$\Gamma_0(N)$$-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])

$$L^{(2)}(E,1)/2!$$ ≈ $$10.3381464196$$

## 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$$(j)_{-}$$
$$2$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$3$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$79$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$211$$ $$1$$ $$I_{1}$$ Non-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 79 211 split nonsplit ordinary ordinary ss ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary nonsplit nonsplit 4 4 2 2 4,2 2 2,2 2 2 2 2 2 2 2 2 2 2 0 0 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 100014b 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 3.3.400056.1 $$\Z/2\Z$$ Not in database
6 $$x^{6}$$ $$\mathstrut -\mathstrut 2 x^{5}$$ $$\mathstrut -\mathstrut 687 x^{4}$$ $$\mathstrut -\mathstrut 5504 x^{3}$$ $$\mathstrut +\mathstrut 24514 x^{2}$$ $$\mathstrut +\mathstrut 129768 x$$ $$\mathstrut -\mathstrut 416184$$ $$\Z/2\Z \times \Z/2\Z$$ 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.