Properties

Label 100011c1
Conductor 100011
Discriminant -106013477073021185065059
j-invariant \( \frac{81228381874428716689043456}{106013477073021185065059} \)
CM no
Rank 1
Torsion Structure \(\mathrm{Trivial}\)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

Minimal Weierstrass equation

magma: E := EllipticCurve([0, -1, 1, 9022524, -11690123677]); // or
magma: E := EllipticCurve("100011c1");
sage: E = EllipticCurve([0, -1, 1, 9022524, -11690123677]) # or
sage: E = EllipticCurve("100011c1")
gp: E = ellinit([0, -1, 1, 9022524, -11690123677]) \\ or
gp: E = ellinit("100011c1")

\( y^2 + y = x^{3} - x^{2} + 9022524 x - 11690123677 \)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(\frac{1837021}{900}, \frac{3331586269}{27000}\right) \)
\(\hat{h}(P)\) ≈  4.52174292299

Integral points

magma: IntegralPoints(E);
sage: E.integral_points()
None

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: \(-106013477073021185065059 \)  =  \(-1 \cdot 3^{7} \cdot 17^{3} \cdot 37^{8} \cdot 53^{2} \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( \frac{81228381874428716689043456}{106013477073021185065059} \)  =  \(2^{12} \cdot 3^{-7} \cdot 17^{-3} \cdot 19^{3} \cdot 37^{-8} \cdot 53^{-2} \cdot 229^{3} \cdot 6221^{3}\)
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: \(4.52174292299\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
Real period: \(0.0565182259796\)
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: \( 48 \)  = \( 1\cdot3\cdot2^{3}\cdot2 \)
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: \(1\) (exact)

Modular invariants

Modular form 100011.2.a.e

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)

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

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

magma: ModularDegree(E);
sage: E.modular_degree()
Modular degree: 10765440
\( \Gamma_0(N) \)-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])

\( L'(E,1) \) ≈ \( 12.2669226405 \)

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\((j)_{-}\)
\(3\) \(1\) \( I_{7} \) Non-split multiplicative 1 1 7 7
\(17\) \(3\) \( I_{3} \) Split multiplicative -1 1 3 3
\(37\) \(8\) \( I_{8} \) Split multiplicative -1 1 8 8
\(53\) \(2\) \( I_{2} \) 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 53
Reduction type ss nonsplit ordinary ordinary ordinary ordinary split ordinary ordinary ordinary ordinary split ordinary ordinary ordinary nonsplit
$\lambda$-invariant(s) 2,17 3 1 1 1 1 2 1 1 1 1 2 1 3 1 1
$\mu$-invariant(s) 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 100011c 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 3.1.204.1 \(\Z/2\Z\) Not in database
6 6.0.2122416.1 \(\Z/2\Z \times \Z/2\Z\) 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.