# Properties

 Label 100011.a1 Conductor 100011 Discriminant -99811078011 j-invariant $$\frac{135353378115584}{99811078011}$$ CM no Rank 1 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([0, 1, 1, 1070, 7408]); // or

magma: E := EllipticCurve("100011e1");

sage: E = EllipticCurve([0, 1, 1, 1070, 7408]) # or

sage: E = EllipticCurve("100011e1")

gp: E = ellinit([0, 1, 1, 1070, 7408]) \\ or

gp: E = ellinit("100011e1")

$$y^2 + y = x^{3} + x^{2} + 1070 x + 7408$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-4, 55\right)$$ $$\hat{h}(P)$$ ≈ 0.215488720298

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-4, 55\right)$$, $$\left(-4, -56\right)$$, $$\left(17, 175\right)$$, $$\left(17, -176\right)$$, $$\left(107, 1165\right)$$, $$\left(107, -1166\right)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$100011$$ = $$3 \cdot 17 \cdot 37 \cdot 53$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-99811078011$$ = $$-1 \cdot 3^{7} \cdot 17 \cdot 37^{3} \cdot 53$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{135353378115584}{99811078011}$$ = $$2^{12} \cdot 3^{-7} \cdot 17^{-1} \cdot 37^{-3} \cdot 53^{-1} \cdot 3209^{3}$$ 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: $$0.215488720298$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.678544484491$$ 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: $$21$$  = $$7\cdot1\cdot3\cdot1$$ 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 100011.2.a.a

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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 121968 $$\Gamma_0(N)$$-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])

$$L'(E,1)$$ ≈ $$3.0705923352$$

## 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$$(j)_{-}$$
$$3$$ $$7$$ $$I_{7}$$ Split multiplicative -1 1 7 7
$$17$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$37$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$53$$ $$1$$ $$I_{1}$$ 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 53 ss split ordinary ss ordinary ordinary split ordinary ss ordinary ordinary split ordinary ordinary ordinary split 2,1 6 1 1,1 3 1 2 1 1,1 1 1 2 1 1 1 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 100011.a 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 3.1.400044.1 $$\Z/2\Z$$ Not in database
6 $$x^{6} + 288 x^{4} + 20736 x^{2} + 3600396$$ $$\Z/2\Z \times \Z/2\Z$$ 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.