Properties

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

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

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -4, -1])
 
gp: E = ellinit([1, 0, 0, -4, -1])
 
magma: E := EllipticCurve([1, 0, 0, -4, -1]);
 

\(y^2+xy=x^3-4x-1\)  Toggle raw display

Mordell-Weil group structure

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

Torsion generators

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

\( \left(-2, 1\right) \), \( \left(5, 8\right) \)  Toggle raw display

Integral points

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

\( \left(-2, 1\right) \), \( \left(-1, 2\right) \), \( \left(-1, -1\right) \), \( \left(2, -1\right) \), \( \left(5, 8\right) \), \( \left(5, -13\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 21 \)  =  $3 \cdot 7$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $3969 $  =  $3^{4} \cdot 7^{2} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{7189057}{3969} \)  =  $3^{-4} \cdot 7^{-2} \cdot 193^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $-0.61894179716900553865896626679\dots$
Stable Faltings height: $-0.61894179716900553865896626679\dots$

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: $3.6089232431079364447472012512\dots$
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: $ 8 $  = $ 2^{2}\cdot2 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $8$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $1$ (exact)
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);
 
Special value: $ L(E,1) $ ≈ $ 0.45111540538849205559340015640263020881 $

Modular invariants

Modular form   21.2.a.a

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 1
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1

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

Galois representations

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 $\ell$-adic Galois representation has maximal image $\GL(2,\Z_\ell)$ for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2Cs 8.48.0.24

$p$-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 7
Reduction type ordinary split nonsplit
$\lambda$-invariant(s) 1 1 0
$\mu$-invariant(s) 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 and 4.
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 less than 24 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
$4$ \(\Q(\sqrt{3}, \sqrt{7})\) \(\Z/2\Z \times \Z/8\Z\) Not in database
$8$ 8.0.49787136.1 \(\Z/4\Z \times \Z/8\Z\) Not in database
$8$ 8.0.9144576.2 \(\Z/2\Z \times \Z/16\Z\) Not in database
$8$ 8.2.425329947.2 \(\Z/2\Z \times \Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/4\Z \times \Z/8\Z\) Not in database
$16$ 16.0.59447875862838378496.4 \(\Z/4\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/16\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/24\Z\) Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.