# Properties

 Label 21a4 Conductor $21$ Discriminant $-63$ j-invariant $$\frac{103823}{63}$$ CM no Rank $0$ Torsion structure $$\Z/{4}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy=x^3+x$$ y^2+xy=x^3+x (homogenize, simplify) $$y^2z+xyz=x^3+xz^2$$ y^2z+xyz=x^3+xz^2 (dehomogenize, simplify) $$y^2=x^3+1269x-3834$$ y^2=x^3+1269x-3834 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 0, 0, 1, 0])

gp: E = ellinit([1, 0, 0, 1, 0])

magma: E := EllipticCurve([1, 0, 0, 1, 0]);

oscar: E = EllipticCurve([1, 0, 0, 1, 0])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z/{4}\Z$$

magma: MordellWeilGroup(E);

## Torsion generators

$$\left(1, 1\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(0, 0\right)$$, $$\left(1, 1\right)$$, $$\left(1, -2\right)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$21$$ = $3 \cdot 7$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $-63$ = $-1 \cdot 3^{2} \cdot 7$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{103823}{63}$$ = $3^{-2} \cdot 7^{-1} \cdot 47^{3}$ comment: j-invariant  sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E);  oscar: j_invariant(E) Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $-0.96551538744897819336758232752\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-0.96551538744897819336758232752\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E)

## BSD invariants

 Analytic rank: $0$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $1$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $3.6089232431079364447472012512\dots$ comment: Real Period  sage: E.period_lattice().omega()  gp: if(E.disc>0,2,1)*E.omega[1]  magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E); Tamagawa product: $2$  = $2\cdot1$ comment: Tamagawa numbers  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E);  oscar: tamagawa_numbers(E) Torsion order: $4$ comment: Torsion order  sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E));  oscar: prod(torsion_structure(E)[1]) Analytic order of Ш: $1$ (exact) comment: Order of Sha  sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Special value: $L(E,1)$ ≈ $0.45111540538849205559340015640$ comment: Special L-value  r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: [r,L1r] = ellanalyticrank(E); L1r/r!  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

## BSD formula

$\displaystyle 0.451115405 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 3.608923 \cdot 1.000000 \cdot 2}{4^2} \approx 0.451115405$

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)

E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;

Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();

omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();

assert r == ar; print("anayltic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))

/* self-contained Magma code snippet for the BSD formula (checks rank, computes analyiic sha) */

E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;

sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);

reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);

assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);

## Modular invariants

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

comment: q-expansion of modular form

sage: E.q_eigenform(20)

\\ actual modular form, use for small N

[mf,F] = mffromell(E)

Ser(mfcoefs(mf,20),q)

\\ or just the series

Ser(ellan(E,20),q)*q

magma: ModularForm(E);

Modular degree: 2
comment: Modular degree

sage: E.modular_degree()

gp: ellmoddegree(E)

magma: ModularDegree(E);

$\Gamma_0(N)$-optimal: no
Manin constant: 2
comment: Manin constant

magma: ManinConstant(E);

## Local data

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

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$3$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$7$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1

comment: Local data

sage: E.local_data()

gp: ellglobalred(E)[5]

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]

## Galois representations

The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2B 16.48.0.27

comment: mod p Galois image

sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

gens = [[5, 4, 332, 333], [1, 16, 0, 1], [125, 16, 304, 21], [15, 2, 238, 323], [305, 16, 74, 207], [321, 16, 320, 17], [1, 16, 92, 213], [1, 0, 16, 1], [160, 5, 243, 322]]

GL(2,Integers(336)).subgroup(gens)

Gens := [[5, 4, 332, 333], [1, 16, 0, 1], [125, 16, 304, 21], [15, 2, 238, 323], [305, 16, 74, 207], [321, 16, 320, 17], [1, 16, 92, 213], [1, 0, 16, 1], [160, 5, 243, 322]];

sub<GL(2,Integers(336))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$336 = 2^{4} \cdot 3 \cdot 7$$, index $192$, genus $1$, and generators

$\left(\begin{array}{rr} 5 & 4 \\ 332 & 333 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 125 & 16 \\ 304 & 21 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 238 & 323 \end{array}\right),\left(\begin{array}{rr} 305 & 16 \\ 74 & 207 \end{array}\right),\left(\begin{array}{rr} 321 & 16 \\ 320 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 92 & 213 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 160 & 5 \\ 243 & 322 \end{array}\right)$.

The torsion field $K:=\Q(E[336])$ is a degree-$12386304$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/336\Z)$.

## $p$-adic regulators

All $p$-adic regulators are identically $1$ since the rank is $0$.

## Iwasawa invariants

$p$ Reduction type $\lambda$-invariant(s) 2 3 7 ord 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

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 4 and 8.
Its isogeny class 21a 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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-7})$$ $$\Z/2\Z \oplus \Z/4\Z$$ 2.0.7.1-63.1-a2 $2$ $$\Q(\sqrt{21})$$ $$\Z/8\Z$$ 2.2.21.1-21.1-b2 $2$ $$\Q(\sqrt{-3})$$ $$\Z/8\Z$$ 2.0.3.1-147.2-a4 $4$ $$\Q(\sqrt{-3}, \sqrt{-7})$$ $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $4$ 4.0.189.1 $$\Z/16\Z$$ Not in database $8$ 8.0.2439569664.6 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.30118144.2 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.4.21956126976.1 $$\Z/16\Z$$ Not in database $8$ 8.0.1750329.1 $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $8$ 8.2.425329947.2 $$\Z/12\Z$$ Not in database $16$ 16.0.5951500145509072896.2 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $16$ 16.0.482071511786234904576.2 $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $16$ deg 16 $$\Z/32\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $16$ deg 16 $$\Z/24\Z$$ Not in database $16$ deg 16 $$\Z/24\Z$$ 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.