# Properties

 Label 100014.f1 Conductor 100014 Discriminant 298640203776 j-invariant $$\frac{1171205436932929}{298640203776}$$ CM no Rank 2 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -2196, -29808]); // or
magma: E := EllipticCurve("100014e1");
sage: E = EllipticCurve([1, 0, 0, -2196, -29808]) # or
sage: E = EllipticCurve("100014e1")
gp: E = ellinit([1, 0, 0, -2196, -29808]) \\ or
gp: E = ellinit("100014e1")

$$y^2 + x y = x^{3} - 2196 x - 29808$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

magma: Generators(E);
sage: E.gens()

 $$P$$ = $$\left(-36, 72\right)$$ $$\left(-24, 108\right)$$ $$\hat{h}(P)$$ ≈ 0.740366497957 0.541792442196

## Integral points

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

$$\left(-36, 72\right)$$, $$\left(-24, 108\right)$$, $$\left(-18, 72\right)$$, $$\left(-16, 44\right)$$, $$\left(54, 72\right)$$, $$\left(72, 396\right)$$, $$\left(96, 756\right)$$, $$\left(216, 2988\right)$$, $$\left(312, 5292\right)$$, $$\left(744, 19884\right)$$, $$\left(2484, 122544\right)$$

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: $$298640203776$$ = $$2^{13} \cdot 3^{7} \cdot 79 \cdot 211$$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j j-invariant: $$\frac{1171205436932929}{298640203776}$$ = $$2^{-13} \cdot 3^{-7} \cdot 23^{3} \cdot 79^{-1} \cdot 211^{-1} \cdot 4583^{3}$$ 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: $$0.272567573791$$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] Real period: $$0.710523308741$$ 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: $$91$$  = $$13\cdot7\cdot1\cdot1$$ 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.f

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)

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

 magma: ModularDegree(E); sage: E.modular_degree() Modular degree: 148512 $$\Gamma_0(N)$$-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])

$$L^{(2)}(E,1)/2!$$ ≈ $$17.6235709091$$

## 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$$(j)_{-}$$
$$2$$ $$13$$ $$I_{13}$$ Split multiplicative -1 1 13 13
$$3$$ $$7$$ $$I_{7}$$ Split multiplicative -1 1 7 7
$$79$$ $$1$$ $$I_{1}$$ 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 split ordinary ordinary ss ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary split nonsplit 4 3 2 2 2,2 2 2 2 2,2 2 2 4 2 2 2 3 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 100014.f 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.