# Properties

 Label 21a5 Conductor $21$ Discriminant $147$ j-invariant $$\frac{53297461115137}{147}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy=x^3-784x-8515$$ y^2+xy=x^3-784x-8515 (homogenize, simplify) $$y^2z+xyz=x^3-784xz^2-8515z^3$$ y^2z+xyz=x^3-784xz^2-8515z^3 (dehomogenize, simplify) $$y^2=x^3-1016091x-394227594$$ y^2=x^3-1016091x-394227594 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 0, 0, -784, -8515])

gp: E = ellinit([1, 0, 0, -784, -8515])

magma: E := EllipticCurve([1, 0, 0, -784, -8515]);

oscar: E = EllipticCurve([1, 0, 0, -784, -8515])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z/{2}\Z$$

magma: MordellWeilGroup(E);

## Torsion generators

$$\left(-\frac{65}{4}, \frac{65}{8}\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

None

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: $147$ = $3 \cdot 7^{2}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{53297461115137}{147}$$ = $3^{-1} \cdot 7^{-2} \cdot 37633^{3}$ 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: $0.074205383390939770758265854672\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $0.074205383390939770758265854672\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E) $abc$ quality: $1.0508747826837916\dots$ Szpiro ratio: $10.381565732764942\dots$

## 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: $0.90223081077698411118680031280\dots$ 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: $2$  = $1\cdot2$ 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: $2$ 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

$\displaystyle 0.451115405 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.902231 \cdot 1.000000 \cdot 2}{2^2} \approx 0.451115405$

# 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("analytic 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: 4
comment: Modular degree

sage: E.modular_degree()

gp: ellmoddegree(E)

magma: ModularDegree(E);

$\Gamma_0(N)$-optimal: no
Manin constant: 1
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$(j)_{-}$
$3$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$7$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

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 16.48.0.48

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 = [[15, 2, 238, 323], [1, 16, 0, 1], [109, 16, 128, 21], [128, 5, 291, 322], [5, 4, 332, 333], [134, 89, 213, 170], [321, 16, 320, 17], [1, 0, 16, 1], [13, 16, 164, 105]]

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

Gens := [[15, 2, 238, 323], [1, 16, 0, 1], [109, 16, 128, 21], [128, 5, 291, 322], [5, 4, 332, 333], [134, 89, 213, 170], [321, 16, 320, 17], [1, 0, 16, 1], [13, 16, 164, 105]];

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

$\left(\begin{array}{rr} 15 & 2 \\ 238 & 323 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 109 & 16 \\ 128 & 21 \end{array}\right),\left(\begin{array}{rr} 128 & 5 \\ 291 & 322 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 332 & 333 \end{array}\right),\left(\begin{array}{rr} 134 & 89 \\ 213 & 170 \end{array}\right),\left(\begin{array}{rr} 321 & 16 \\ 320 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 13 & 16 \\ 164 & 105 \end{array}\right)$.

Input positive integer $m$ to see the generators of the reduction of $H$ to $\mathrm{GL}_2(\Z/m\Z)$:

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 $\GL_2(\Z/336\Z)$.

## 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.

## Twists

This elliptic curve is its own minimal quadratic twist.

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

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{3})$$ $$\Z/2\Z \oplus \Z/2\Z$$ 2.2.12.1-147.1-a7 $2$ $$\Q(\sqrt{-3})$$ $$\Z/4\Z$$ 2.0.3.1-147.2-a8 $2$ $$\Q(\sqrt{-1})$$ $$\Z/4\Z$$ 2.0.4.1-441.1-a6 $4$ $$\Q(\zeta_{12})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ 4.2.338688.3 $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $4$ 4.0.432.1 $$\Z/8\Z$$ Not in database $4$ $$\Q(i, \sqrt{42})$$ $$\Z/8\Z$$ Not in database $4$ $$\Q(i, \sqrt{14})$$ $$\Z/8\Z$$ Not in database $8$ 8.0.114709561344.25 $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.0.2985984.1 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.0.12745506816.9 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $8$ 8.0.7023034368.6 $$\Z/16\Z$$ Not in database $8$ 8.2.425329947.2 $$\Z/6\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $16$ deg 16 $$\Z/16\Z$$ Not in database $16$ deg 16 $$\Z/16\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/16\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $16$ deg 16 $$\Z/12\Z$$ Not in database $16$ deg 16 $$\Z/12\Z$$ 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.

## Iwasawa invariants

$p$ Reduction type $\lambda$-invariant(s) 2 3 7 ord split nonsplit 1 1 0 2 0 0

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

## $p$-adic regulators

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