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()

\(P\) =  \( \left(-13, 38\right) \)\( \left(-6, 45\right) \)
\(\hat{h}(P)\) ≈  0.5335189098070.607544056742

Integral points

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

\( \left(-15, 26\right) \), \( \left(-13, 38\right) \), \( \left(-6, 45\right) \), \( \left(-2, 38\right) \), \( \left(2, 26\right) \), \( \left(7, 6\right) \), \( \left(8, 3\right) \), \( \left(9, 5\right) \), \( \left(13, 26\right) \), \( \left(15, 38\right) \), \( \left(20, 71\right) \), \( \left(42, 258\right) \), \( \left(64, 500\right) \), \( \left(85, 773\right) \), \( \left(163, 2073\right) \), \( \left(449, 9509\right) \)

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

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 form 1001.2.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

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

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ss ss ordinary nonsplit split nonsplit ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary
$\lambda$-invariant(s) 3,4 2,8 2 2 3 4 2 2 2 2 2 2 2,2 2 2
$\mu$-invariant(s) 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.