# Properties

 Label 126.b3 Conductor $126$ Discriminant $686128968$ j-invariant $$\frac{4956477625}{941192}$$ CM no Rank $0$ Torsion structure $$\Z/{6}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2+xy+y=x^3-x^2-320x+1883$$ y^2+xy+y=x^3-x^2-320x+1883 (homogenize, simplify) $$y^2z+xyz+yz^2=x^3-x^2z-320xz^2+1883z^3$$ y^2z+xyz+yz^2=x^3-x^2z-320xz^2+1883z^3 (dehomogenize, simplify) $$y^2=x^3-5115x+115414$$ y^2=x^3-5115x+115414 (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, -1, 1, -320, 1883])

gp: E = ellinit([1, -1, 1, -320, 1883])

magma: E := EllipticCurve([1, -1, 1, -320, 1883]);

oscar: E = EllipticCurve([1, -1, 1, -320, 1883])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

$$\Z/{6}\Z$$

magma: MordellWeilGroup(E);

## Torsion generators

$$\left(-9, 67\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(-9, 67\right)$$, $$\left(-9, -59\right)$$, $$\left(19, 39\right)$$, $$\left(19, -59\right)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$126$$ = $2 \cdot 3^{2} \cdot 7$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $686128968$ = $2^{3} \cdot 3^{6} \cdot 7^{6}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$\frac{4956477625}{941192}$$ = $2^{-3} \cdot 5^{3} \cdot 7^{-6} \cdot 11^{3} \cdot 31^{3}$ 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: $0.41310098042308445758078235484\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-0.13620516391097038811684026362\dots$ 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: $1.5305454480783489375703797074\dots$ 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: $36$  = $3\cdot2\cdot( 2 \cdot 3 )$ 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: $6$ 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: $L(E,1)$ ≈ $1.5305454480783489375703797074$ 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

$\displaystyle 1.530545448 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.530545 \cdot 1.000000 \cdot 36}{6^2} \approx 1.530545448$

# 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

$$q + q^{2} + q^{4} + q^{7} + q^{8} - 4 q^{13} + q^{14} + q^{16} - 6 q^{17} + 2 q^{19} + O(q^{20})$$

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: 48
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 not semistable. There are 3 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$3$ $2$ $I_0^{*}$ Additive -1 2 6 0
$7$ $6$ $I_{6}$ Split multiplicative -1 1 6 6

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 8.6.0.6
$3$ 3Cs.1.1 3.24.0.1

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 = [[1, 36, 0, 1], [325, 276, 18, 163], [1, 0, 36, 1], [251, 468, 0, 475], [469, 36, 468, 37], [445, 36, 174, 277], [1, 6, 6, 37], [10, 3, 249, 352], [19, 24, 432, 307]]

GL(2,Integers(504)).subgroup(gens)

Gens := [[1, 36, 0, 1], [325, 276, 18, 163], [1, 0, 36, 1], [251, 468, 0, 475], [469, 36, 468, 37], [445, 36, 174, 277], [1, 6, 6, 37], [10, 3, 249, 352], [19, 24, 432, 307]];

sub<GL(2,Integers(504))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level $$504 = 2^{3} \cdot 3^{2} \cdot 7$$, index $864$, genus $21$, and generators

$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 325 & 276 \\ 18 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 251 & 468 \\ 0 & 475 \end{array}\right),\left(\begin{array}{rr} 469 & 36 \\ 468 & 37 \end{array}\right),\left(\begin{array}{rr} 445 & 36 \\ 174 & 277 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 10 & 3 \\ 249 & 352 \end{array}\right),\left(\begin{array}{rr} 19 & 24 \\ 432 & 307 \end{array}\right)$.

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

The torsion field $K:=\Q(E[504])$ is a degree-$13934592$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/504\Z)$.

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 126.b consists of 6 curves linked by isogenies of degrees dividing 18.

## Twists

The minimal quadratic twist of this elliptic curve is 14.a3, its twist by $-3$.

## 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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{2})$$ $$\Z/2\Z \oplus \Z/6\Z$$ Not in database $2$ $$\Q(\sqrt{-3})$$ $$\Z/3\Z \oplus \Z/6\Z$$ 2.0.3.1-196.2-a4 $4$ 4.0.14112.1 $$\Z/12\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{-3})$$ $$\Z/6\Z \oplus \Z/6\Z$$ Not in database $8$ 8.4.16647192576.5 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $8$ 8.0.12745506816.13 $$\Z/2\Z \oplus \Z/12\Z$$ Not in database $8$ 8.0.199148544.2 $$\Z/3\Z \oplus \Z/12\Z$$ Not in database $9$ 9.3.12004512424128.4 $$\Z/18\Z$$ Not in database $16$ deg 16 $$\Z/24\Z$$ Not in database $16$ deg 16 $$\Z/6\Z \oplus \Z/12\Z$$ Not in database $16$ 16.0.162447943996702457856.4 $$\Z/6\Z \oplus \Z/12\Z$$ Not in database $18$ 18.0.3674701490222020866048.1 $$\Z/3\Z \oplus \Z/18\Z$$ Not in database $18$ 18.0.432324955623130532869681152.2 $$\Z/3\Z \oplus \Z/18\Z$$ Not in database $18$ 18.0.144784752906623254803.2 $$\Z/3\Z \oplus \Z/18\Z$$ Not in database $18$ 18.6.302217048444986481089572522426368.3 $$\Z/2\Z \oplus \Z/18\Z$$ 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 split add split 2 - 1 0 - 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

An entry - indicates that the invariants are not computed because the reduction is additive.

## $p$-adic regulators

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