Properties

 Label 1001.a1 Conductor 1001 Discriminant -11022011 j-invariant $$-\frac{871531204608}{11022011}$$ CM no Rank 2 Torsion Structure $$\mathrm{Trivial}$$

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Minimal Weierstrass equation

magma: E := EllipticCurve([0, 0, 1, -199, 1092]); // or

magma: E := EllipticCurve("1001c1");

sage: E = EllipticCurve([0, 0, 1, -199, 1092]) # or

sage: E = EllipticCurve("1001c1")

gp: E = ellinit([0, 0, 1, -199, 1092]) \\ or

gp: E = ellinit("1001c1")

$$y^2 + y = x^{3} - 199 x + 1092$$

Mordell-Weil group structure

$$\Z^2$$

Infinite order Mordell-Weil generators and heights

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-13, 38\right)$$ $$\left(-6, 45\right)$$ $$\hat{h}(P)$$ ≈ 0.533518909807 0.607544056742

Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-15, 26\right)$$, $$\left(-15, -27\right)$$, $$\left(-13, 38\right)$$, $$\left(-13, -39\right)$$, $$\left(-6, 45\right)$$, $$\left(-6, -46\right)$$, $$\left(-2, 38\right)$$, $$\left(-2, -39\right)$$, $$\left(2, 26\right)$$, $$\left(2, -27\right)$$, $$\left(7, 6\right)$$, $$\left(7, -7\right)$$, $$\left(8, 3\right)$$, $$\left(8, -4\right)$$, $$\left(9, 5\right)$$, $$\left(9, -6\right)$$, $$\left(13, 26\right)$$, $$\left(13, -27\right)$$, $$\left(15, 38\right)$$, $$\left(15, -39\right)$$, $$\left(20, 71\right)$$, $$\left(20, -72\right)$$, $$\left(42, 258\right)$$, $$\left(42, -259\right)$$, $$\left(64, 500\right)$$, $$\left(64, -501\right)$$, $$\left(85, 773\right)$$, $$\left(85, -774\right)$$, $$\left(163, 2073\right)$$, $$\left(163, -2074\right)$$, $$\left(449, 9509\right)$$, $$\left(449, -9510\right)$$

Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$1001$$ = $$7 \cdot 11 \cdot 13$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-11022011$$ = $$-1 \cdot 7^{2} \cdot 11^{3} \cdot 13^{2}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{871531204608}{11022011}$$ = $$-1 \cdot 2^{12} \cdot 3^{3} \cdot 7^{-2} \cdot 11^{-3} \cdot 13^{-2} \cdot 199^{3}$$ 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: $$0.0360107457562$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$2.28178753509$$ 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: $$12$$  = $$2\cdot3\cdot2$$ 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 form1001.2.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)

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

For more coefficients, see the Downloads section to the right.

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 1008 $$\Gamma_0(N)$$-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])

$$L^{(2)}(E,1)/2!$$ ≈ $$0.98602644955$$

Local data

This elliptic curve is semistable.

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$$(j)_{-}$$
$$7$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$11$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$13$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2

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 ss ss ordinary nonsplit split nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary 3,4 2,8 2 2 3 4 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

Isogenies

This curve has no rational isogenies. Its isogeny class 1001.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 3.1.44.1 $$\Z/2\Z$$ Not in database
6 6.0.21296.1 $$\Z/2\Z \times \Z/2\Z$$ 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.