# Properties

 Label 15.a8 Conductor $15$ Discriminant $-3515625$ j-invariant $\frac{4733169839}{3515625}$ CM no Rank $0$ Torsion Structure $\Z/{8}\Z$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 1, 1, 35, -28]); // or
magma: E := EllipticCurve("15a4");
sage: E = EllipticCurve([1, 1, 1, 35, -28]) # or
sage: E = EllipticCurve("15a4")
gp: E = ellinit([1, 1, 1, 35, -28]) \\ or
gp: E = ellinit("15a4")

$y^2 + x y + y = x^{3} + x^{2} + 35 x - 28$

## Mordell-Weil group structure

$\Z/{8}\Z$

## Torsion generators

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

$\left(2, 6\right)$

## Integral points

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

$\left(2, 6\right)$, $\left(7, 21\right)$, $\left(32, 171\right)$

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] $N$ = $15$ = $3 \cdot 5$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc $\Delta$ = $-3515625$ = $-1 \cdot 3^{2} \cdot 5^{8}$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j $j$ = $\frac{4733169839}{3515625}$ = $3^{-2} \cdot 5^{-8} \cdot 23^{3} \cdot 73^{3}$ $\text{End} (E)$ = $\Z$ (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] $\Omega$ ≈ $1.40060304233$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] $\prod_p c_p$ = $16$  = $2\cdot2^{3}$ magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp: elltors(E)[1] $\#E_{\text{tor}}$ = $8$ magma: MordellWeilShaInformation(E); sage: E.sha().an_numerical() Ш$_{\text{an}}$ = $1$ (exact)

## Modular invariants

### Modular form15.2.1.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)

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

### Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
2 : 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])

$L(E,1)$ ≈ $0.350150760583$

## 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$(j)_{-}$
$3$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$5$ $8$ $I_{8}$ Split multiplicative -1 1 8 8

## Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X195l.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 3 \\ 0 & 1 \end{array}\right)$ and has index 96.

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
$2$ B

## $p$-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 5 ordinary nonsplit split 0 0 1 2 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4, 8 and 16.
Its isogeny class 15.a consists of 8 curves linked by isogenies of degrees dividing 16.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{8}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $\Q(\sqrt{-1})$ $\Z/2\Z \times \Z/8\Z$ 2.0.4.1-225.2-a5
4 4.2.3600.1 $\Z/16\Z$ Not in database
$\Q(i, \sqrt{6})$ $\Z/4\Z \times \Z/8\Z$ 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.