# Properties

 Label 21.a5 Conductor $21$ Discriminant $3969$ j-invariant $\frac{7189057}{3969}$ CM no Rank $0$ Torsion Structure $\Z/{2}\Z \times \Z/{4}\Z$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

This is a model for the modular curve $X_0(21)$.

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -4, -1]); // or
magma: E := EllipticCurve("21a1");
sage: E = EllipticCurve([1, 0, 0, -4, -1]) # or
sage: E = EllipticCurve("21a1")
gp: E = ellinit([1, 0, 0, -4, -1]) \\ or
gp: E = ellinit("21a1")

$y^2 + x y = x^{3} - 4 x - 1$

## Mordell-Weil group structure

$\Z/{2}\Z \times \Z/{4}\Z$

## Torsion generators

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

$\left(-2, 1\right)$, $\left(5, 8\right)$

## Integral points

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

$\left(-2, 1\right)$, $\left(-1, 2\right)$, $\left(2, -1\right)$, $\left(5, 8\right)$

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

## Invariants

 magma: Conductor(E); sage: E.conductor().factor() gp: ellglobalred(E)[1] $N$ = $21$ = $3 \cdot 7$ magma: Discriminant(E); sage: E.discriminant().factor() gp: E.disc $\Delta$ = $3969$ = $3^{4} \cdot 7^{2}$ magma: jInvariant(E); sage: E.j_invariant().factor() gp: E.j $j$ = $\frac{7189057}{3969}$ = $3^{-4} \cdot 7^{-2} \cdot 193^{3}$ $\text{End} (E)$ = $\Z$ (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] $\Omega$ ≈ $3.60892324311$ magma: TamagawaNumbers(E); sage: E.tamagawa_numbers() gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] $\prod_p c_p$ = $8$  = $2^{2}\cdot2$ 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 form21.2.1.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)

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

### Modular degree and optimality

magma: ModularDegree(E);
sage: E.modular_degree()
1 : curve is $\Gamma_0(N)$-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])

$L(E,1)$ ≈ $0.451115405388$

## 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$(j)_{-}$
$3$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$7$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

## 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 X98e.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 2 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 4 & 5 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right)$ and has index 48.

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
$2$ Cs

## $p$-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 7 ordinary split nonsplit 1 1 0 0 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 and 8.
Its isogeny class 21.a consists of 6 curves linked by isogenies of degrees dividing 8.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{4}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $\Q(\sqrt{-3})$ $\Z/2\Z \times \Z/8\Z$ 2.0.3.1-147.2-a5
4 $\Q(i, \sqrt{7})$ $\Z/4\Z \times \Z/4\Z$ Not in database
$\Q(\sqrt{3}, \sqrt{7})$ $\Z/2\Z \times \Z/8\Z$ 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.