Properties

Label 26.b1
Conductor $26$
Discriminant $-125497034$
j-invariant \( -\frac{1064019559329}{125497034} \)
CM no
Rank $0$
Torsion structure trivial

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Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 1, -213, -1257])
 
gp: E = ellinit([1, -1, 1, -213, -1257])
 
magma: E := EllipticCurve([1, -1, 1, -213, -1257]);
 

\(y^2+xy+y=x^3-x^2-213x-1257\)  Toggle raw display

Mordell-Weil group structure

trivial

Integral points

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

\(\)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 26 \)  =  \(2 \cdot 13\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-125497034 \)  =  \(-1 \cdot 2 \cdot 13^{7} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{1064019559329}{125497034} \)  =  \(-1 \cdot 2^{-1} \cdot 3^{3} \cdot 13^{-7} \cdot 41^{3} \cdot 83^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: \(0.28979452610338456519600707002\dots\)
Stable Faltings height: \(0.28979452610338456519600707002\dots\)

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: \(0\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(1\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.62096534954905546637586267270\dots\)
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: \( 1 \)
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   26.2.a.b

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 14
\( \Gamma_0(N) \)-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);
 

\( L(E,1) \) ≈ \( 0.62096534954905546637586267270088986448 \)

Local data

This elliptic curve is semistable. There are 2 primes of bad reduction:

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
\(13\) \(1\) \(I_{7}\) Non-split multiplicative 1 1 7 7

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
\(7\) B.1.3

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) 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 13
Reduction type split ss ordinary ordinary nonsplit
$\lambda$-invariant(s) 1 0,0 0 4 0
$\mu$-invariant(s) 0 0,0 0 1 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 26.b consists of 2 curves linked by isogenies of degree 7.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 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
$3$ 3.1.104.1 \(\Z/2\Z\) Not in database
$6$ 6.0.1124864.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
$6$ \(\Q(\zeta_{7})\) \(\Z/7\Z\) Not in database
$7$ 7.1.52706752.1 \(\Z/7\Z\) Not in database
$8$ 8.2.999406512.1 \(\Z/3\Z\) Not in database
$12$ 12.2.8421963387109376.7 \(\Z/4\Z\) Not in database
$18$ 18.0.6007179870010464504905728.1 \(\Z/14\Z\) Not in database
$21$ 21.1.301059380309170415238823998755176448.1 \(\Z/14\Z\) Not in database

We only show fields where the torsion growth is primitive.