Properties

Label 100014f2
Conductor 100014
Discriminant 2250945132306174
j-invariant \( \frac{72918170522696196433}{2250945132306174} \)
CM no
Rank 1
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -87037, -9623389]); // or
magma: E := EllipticCurve("100014f2");
sage: E = EllipticCurve([1, 0, 0, -87037, -9623389]) # or
sage: E = EllipticCurve("100014f2")
gp: E = ellinit([1, 0, 0, -87037, -9623389]) \\ or
gp: E = ellinit("100014f2")

\( y^2 + x y = x^{3} - 87037 x - 9623389 \)

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

magma: Generators(E);
sage: E.gens()

\(P\) =  \( \left(\frac{4595}{4}, -\frac{304637}{8}\right) \)
\(\hat{h}(P)\) ≈  0.771181858965

Integral points

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

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: \(2250945132306174 \)  =  \(2 \cdot 3^{5} \cdot 79^{3} \cdot 211^{3} \)
magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
j-invariant: \( \frac{72918170522696196433}{2250945132306174} \)  =  \(2^{-1} \cdot 3^{-5} \cdot 19^{3} \cdot 31^{3} \cdot 41^{3} \cdot 79^{-3} \cdot 173^{3} \cdot 211^{-3}\)
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: \(0.771181858965\)
magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
Real period: \(0.278480078802\)
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: \( 45 \)  = \( 1\cdot5\cdot3\cdot3 \)
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 100014.2.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)

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

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

magma: ModularDegree(E);
sage: E.modular_degree()
Modular degree: 1213920
\( \Gamma_0(N) \)-optimal: no
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])

\( L'(E,1) \) ≈ \( 9.66414531849 \)

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\((j)_{-}\)
\(2\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1
\(3\) \(5\) \( I_{5} \) Split multiplicative -1 1 5 5
\(79\) \(3\) \( I_{3} \) Split multiplicative -1 1 3 3
\(211\) \(3\) \( I_{3} \) Split multiplicative -1 1 3 3

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 \( p \) except those listed.

prime Image of Galois representation
\(3\) B.1.2

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 79 211
Reduction type split split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ss ordinary ordinary split split
$\lambda$-invariant(s) 2 4 1 3 1 1 1 1 1 1 1 1 1,1 1 1 2 2
$\mu$-invariant(s) 0 1 0 0 0 0 0 0 0 0 0 0 0,0 0 0 0 0

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 100014f consists of 2 curves linked by isogenies of degree 3.

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
2 \(\Q(\sqrt{-3}) \) \(\Z/3\Z\) Not in database
3 3.3.400056.1 \(\Z/2\Z\) Not in database
3.1.972.1 \(\Z/3\Z\) Not in database
6 6.0.2834352.1 \(\Z/3\Z \times \Z/3\Z\) Not in database
\(x^{6} \) \(\mathstrut -\mathstrut 3 x^{5} \) \(\mathstrut +\mathstrut 130 x^{4} \) \(\mathstrut -\mathstrut 255 x^{3} \) \(\mathstrut +\mathstrut 4684 x^{2} \) \(\mathstrut -\mathstrut 4557 x \) \(\mathstrut +\mathstrut 32727 \) \(\Z/6\Z\) Not in database
\(x^{6} \) \(\mathstrut -\mathstrut 2 x^{5} \) \(\mathstrut -\mathstrut 687 x^{4} \) \(\mathstrut -\mathstrut 5504 x^{3} \) \(\mathstrut +\mathstrut 24514 x^{2} \) \(\mathstrut +\mathstrut 129768 x \) \(\mathstrut -\mathstrut 416184 \) \(\Z/2\Z \times \Z/2\Z\) 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.