# Properties

 Label 26.a1 Conductor $26$ Discriminant $-6656$ j-invariant $-\frac{10730978619193}{6656}$ CM no Rank $0$ Torsion Structure $\mathrm{Trivial}$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 1, -460, -3830]); // or
magma: E := EllipticCurve("26a2");
sage: E = EllipticCurve([1, 0, 1, -460, -3830]) # or
sage: E = EllipticCurve("26a2")
gp: E = ellinit([1, 0, 1, -460, -3830]) \\ or
gp: E = ellinit("26a2")

$y^2 + x y + y = x^{3} - 460 x - 3830$

Trivial

## Integral points

magma: IntegralPoints(E);
sage: E.integral_points()
None

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] $N$ = $26$ = $2 \cdot 13$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc $\Delta$ = $-6656$ = $-1 \cdot 2^{9} \cdot 13$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j $j$ = $-\frac{10730978619193}{6656}$ = $-1 \cdot 2^{-9} \cdot 7^{3} \cdot 13^{-1} \cdot 23^{3} \cdot 137^{3}$ $\text{End} (E)$ = $\Z$ (no Complex Multiplication) $\text{ST} (E)$ = $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E); sage: E.rank() $r$ = $0$ magma: Regulator(E); sage: E.regulator() $\text{Reg}$ = $1$ magma: RealPeriod(E); sage: E.period_lattice().omega() gp: E.omega[1] $\Omega$ ≈ $0.515576651277$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] $\prod_p c_p$ = $1$  = $1\cdot1$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] $\#E_{\text{tor}}$ = $1$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Ш$_{\text{an}}$ = $1$ (exact)

## Modular invariants

### Modular form26.2.1.a

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

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

### Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
6 : curve is not $\Gamma_0(N)$-optimal

### Special L-value attached to the curve

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])

$L(E,1)$ ≈ $0.515576651277$

## 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$(j)_{-}$
$2$ $1$ $I_{9}$ Non-split multiplicative 1 1 9 9
$13$ $1$ $I_{1}$ 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
$3$ 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

$p$ Reduction type $\lambda$-invariant(s) 2 3 13 nonsplit ordinary split 1 0 1 0 2 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.
Its isogeny class 26.a 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 $\Q(\sqrt{-3})$ $\Z/3\Z$ 2.0.3.1-676.2-b1
3 3.1.104.1 $\Z/2\Z$ Not in database
3.1.4563.1 $\Z/3\Z$ Not in database
6 6.0.62462907.1 $\Z/3\Z \times \Z/3\Z$ Not in database
6.0.292032.1 $\Z/6\Z$ Not in database
6.0.562166163.2 $\Z/9\Z$ Not in database
6.0.3326427.2 $\Z/9\Z$ Not in database
6.0.1124864.1 $\Z/2\Z \times \Z/2\Z$ 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.