Properties

Label 714e1
Conductor $714$
Discriminant $-34349180544$
j-invariant \( -\frac{1184052061112257}{34349180544} \)
CM no
Rank $0$
Torsion structure trivial

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, -2204, -41731]) # or
 
sage: E = EllipticCurve("714e1")
 
gp: E = ellinit([1, 1, 1, -2204, -41731]) \\ or
 
gp: E = ellinit("714e1")
 
magma: E := EllipticCurve([1, 1, 1, -2204, -41731]); // or
 
magma: E := EllipticCurve("714e1");
 

\( y^2 + x y + y = x^{3} + x^{2} - 2204 x - 41731 \)

Mordell-Weil group structure

trivial

Integral points

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

None

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 714 \)  =  \(2 \cdot 3 \cdot 7 \cdot 17\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-34349180544 \)  =  \(-1 \cdot 2^{7} \cdot 3^{3} \cdot 7 \cdot 17^{5} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{1184052061112257}{34349180544} \)  =  \(-1 \cdot 2^{-7} \cdot 3^{-3} \cdot 7^{-1} \cdot 17^{-5} \cdot 67^{3} \cdot 1579^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Rank: \(0\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(1\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.347794905515\)
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: \( 7 \)  = \( 7\cdot1\cdot1\cdot1 \)
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: \(1\) (exact)

Modular invariants

Modular form   714.2.a.g

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 840
\( \Gamma_0(N) \)-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);
 

\( L(E,1) \) ≈ \( 2.4345643386 \)

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\((j)_{-}\)
\(2\) \(7\) \( I_{7} \) Split multiplicative -1 1 7 7
\(3\) \(1\) \( I_{3} \) Non-split multiplicative 1 1 3 3
\(7\) \(1\) \( I_{1} \) Non-split multiplicative 1 1 1 1
\(17\) \(1\) \( I_{5} \) Non-split multiplicative 1 1 5 5

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]
 

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type split nonsplit ordinary nonsplit ordinary ordinary nonsplit ordinary ordinary ordinary ss ordinary ordinary ordinary ss
$\lambda$-invariant(s) 1 0 0 2 2 2 0 0 0 0 0,0 0 0 0 0,0
$\mu$-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 714e 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
$3$ 3.1.2856.1 \(\Z/2\Z\) Not in database
$6$ 6.0.23295638016.1 \(\Z/2\Z \times \Z/2\Z\) 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.