Properties

Label 714.i1
Conductor $714$
Discriminant $-2.441\times 10^{14}$
j-invariant \( -\frac{6150311179917589675873}{244053849830826} \)
CM no
Rank $0$
Torsion structure trivial

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -381702, -90803346]) # or
 
sage: E = EllipticCurve("714i3")
 
gp: E = ellinit([1, 0, 0, -381702, -90803346]) \\ or
 
gp: E = ellinit("714i3")
 
magma: E := EllipticCurve([1, 0, 0, -381702, -90803346]); // or
 
magma: E := EllipticCurve("714i3");
 

\( y^2 + x y = x^{3} - 381702 x - 90803346 \)

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: \(-244053849830826 \)  =  \(-1 \cdot 2 \cdot 3 \cdot 7^{3} \cdot 17^{9} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{6150311179917589675873}{244053849830826} \)  =  \(-1 \cdot 2^{-1} \cdot 3^{-1} \cdot 7^{-3} \cdot 17^{-9} \cdot 37^{3} \cdot 495181^{3}\)
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: \(0.0960367401912\)
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: \( 3 \)  = \( 1\cdot1\cdot3\cdot1 \)
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: \(9\) (exact)

Modular invariants

Modular form   714.2.a.i

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 9720
\( \Gamma_0(N) \)-optimal: no
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);
 

\( L(E,1) \) ≈ \( 2.59299198516 \)

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\((j)_{-}\)
\(2\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1
\(3\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1
\(7\) \(3\) \( I_{3} \) Split multiplicative -1 1 3 3
\(17\) \(1\) \( I_{9} \) Non-split multiplicative 1 1 9 9

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 \( p \) 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]
 

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

Iwasawa invariants

$p$ 2 3 7 17
Reduction type split split split nonsplit
$\lambda$-invariant(s) 1 3 1 0
$\mu$-invariant(s) 0 2 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3 and 9.
Its isogeny class 714.i consists of 3 curves linked by isogenies of degrees dividing 9.

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
$3$ 3.1.972.2 \(\Z/3\Z\) Not in database
$3$ 3.1.2856.1 \(\Z/2\Z\) Not in database
$6$ \(\Q(\zeta_{9})\) \(\Z/9\Z\) Not in database
$6$ 6.0.2834352.2 \(\Z/3\Z \times \Z/3\Z\) Not in database
$6$ 6.0.2834352.4 \(\Z/9\Z\) Not in database
$6$ 6.0.24470208.1 \(\Z/6\Z\) 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.