Properties

Label 100014e1
Conductor 100014
Discriminant 298640203776
j-invariant \( \frac{1171205436932929}{298640203776} \)
CM no
Rank 2
Torsion Structure \(\mathrm{Trivial}\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -2196, -29808]) # or
 
sage: E = EllipticCurve("100014e1")
 
gp: E = ellinit([1, 0, 0, -2196, -29808]) \\ or
 
gp: E = ellinit("100014e1")
 
magma: E := EllipticCurve([1, 0, 0, -2196, -29808]); // or
 
magma: E := EllipticCurve("100014e1");
 

\( y^2 + x y = x^{3} - 2196 x - 29808 \)

Mordell-Weil group structure

\(\Z^2\)

Infinite order Mordell-Weil generators and heights

sage: E.gens()
 
magma: Generators(E);
 

\(P\) =  \( \left(-36, 72\right) \)\( \left(-24, 108\right) \)
\(\hat{h}(P)\) ≈  0.74036649795733710.5417924421958548

Integral points

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

\( \left(-36, 72\right) \), \( \left(-36, -36\right) \), \( \left(-24, 108\right) \), \( \left(-24, -84\right) \), \( \left(-18, 72\right) \), \( \left(-18, -54\right) \), \( \left(-16, 44\right) \), \( \left(-16, -28\right) \), \( \left(54, 72\right) \), \( \left(54, -126\right) \), \( \left(72, 396\right) \), \( \left(72, -468\right) \), \( \left(96, 756\right) \), \( \left(96, -852\right) \), \( \left(216, 2988\right) \), \( \left(216, -3204\right) \), \( \left(312, 5292\right) \), \( \left(312, -5604\right) \), \( \left(744, 19884\right) \), \( \left(744, -20628\right) \), \( \left(2484, 122544\right) \), \( \left(2484, -125028\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 100014 \)  =  \(2 \cdot 3 \cdot 79 \cdot 211\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(298640203776 \)  =  \(2^{13} \cdot 3^{7} \cdot 79 \cdot 211 \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{1171205436932929}{298640203776} \)  =  \(2^{-13} \cdot 3^{-7} \cdot 23^{3} \cdot 79^{-1} \cdot 211^{-1} \cdot 4583^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Rank: \(2\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(0.272567573791\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.710523308741\)
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
Tamagawa product: \( 91 \)  = \( 13\cdot7\cdot1\cdot1 \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(1\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (rounded)

Modular invariants

Modular form 100014.2.a.f

sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 
magma: ModularForm(E);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 148512
\( \Gamma_0(N) \)-optimal: yes
Manin constant: 1

Special L-value

sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 

\( L^{(2)}(E,1)/2! \) ≈ \( 17.6235709091 \)

Local data

This elliptic curve is semistable.

sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(13\) \( I_{13} \) Split multiplicative -1 1 13 13
\(3\) \(7\) \( I_{7} \) Split multiplicative -1 1 7 7
\(79\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1
\(211\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1

Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

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 79 211
Reduction type split split ordinary ordinary ss ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary split nonsplit
$\lambda$-invariant(s) 4 3 2 2 2,2 2 2 2 2,2 2 2 4 2 2 2 3 2
$\mu$-invariant(s) 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 100014e 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 3.3.400056.1 \(\Z/2\Z\) Not in database
6 \( x^{6} - 2 x^{5} - 687 x^{4} - 5504 x^{3} + 24514 x^{2} + 129768 x - 416184 \) \(\Z/2\Z \times \Z/2\Z\) 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.