Properties

Label 678d1
Conductor 678
Discriminant -4048994304
j-invariant \( -\frac{506814405937489}{4048994304} \)
CM no
Rank 0
Torsion Structure \(\Z/{7}\Z\)

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Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -1661, 26097]); // or
 
magma: E := EllipticCurve("678d1");
 
sage: E = EllipticCurve([1, 0, 0, -1661, 26097]) # or
 
sage: E = EllipticCurve("678d1")
 
gp: E = ellinit([1, 0, 0, -1661, 26097]) \\ or
 
gp: E = ellinit("678d1")
 

\( y^2 + x y = x^{3} - 1661 x + 26097 \)

Mordell-Weil group structure

\(\Z/{7}\Z\)

Torsion generators

magma: TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 

\( \left(-14, 223\right) \)

Integral points

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

\( \left(-14, 223\right) \), \( \left(22, 7\right) \), \( \left(34, 79\right) \)

Note: only one of each pair $\pm P$ is listed.

Invariants

magma: Conductor(E);
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
Conductor: \( 678 \)  =  \(2 \cdot 3 \cdot 113\)
magma: Discriminant(E);
 
sage: E.discriminant().factor()
 
gp: E.disc
 
Discriminant: \(-4048994304 \)  =  \(-1 \cdot 2^{14} \cdot 3^{7} \cdot 113 \)
magma: jInvariant(E);
 
sage: E.j_invariant().factor()
 
gp: E.j
 
j-invariant: \( -\frac{506814405937489}{4048994304} \)  =  \(-1 \cdot 2^{-14} \cdot 3^{-7} \cdot 13^{3} \cdot 113^{-1} \cdot 6133^{3}\)
Endomorphism ring: \(\Z\)   (no Complex Multiplication)
Sato-Tate Group: $\mathrm{SU}(2)$

BSD invariants

magma: Rank(E);
 
sage: E.rank()
 
Rank: \(0\)
magma: Regulator(E);
 
sage: E.regulator()
 
Regulator: \(1\)
magma: RealPeriod(E);
 
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
Real period: \(1.39704573143\)
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: \( 98 \)  = \( ( 2 \cdot 7 )\cdot7\cdot1 \)
magma: Order(TorsionSubgroup(E));
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
Torsion order: \(7\)
magma: MordellWeilShaInformation(E);
 
sage: E.sha().an_numerical()
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form 678.2.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)
 

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

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

magma: ModularDegree(E);
 
sage: E.modular_degree()
 
Modular degree: 504
\( \Gamma_0(N) \)-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])
 

\( L(E,1) \) ≈ \( 2.79409146286 \)

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\((j)_{-}\)
\(2\) \(14\) \( I_{14} \) Split multiplicative -1 1 14 14
\(3\) \(7\) \( I_{7} \) Split multiplicative -1 1 7 7
\(113\) \(1\) \( I_{1} \) Split multiplicative -1 1 1 1

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
\(7\) B.1.1

$p$-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 7 113
Reduction type split split ordinary ordinary split
$\lambda$-invariant(s) 1 1 0 2 1
$\mu$-invariant(s) 0 0 0 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 7.
Its isogeny class 678d consists of 2 curves linked by isogenies of degree 7.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{7}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.339.1 \(\Z/14\Z\) Not in database
6 6.0.38958219.1 \(\Z/2\Z \times \Z/14\Z\) 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.