# Properties

 Label 576.c4 Conductor $576$ Discriminant $-46656$ j-invariant $$1728$$ CM yes ($$D=-4$$) Rank $1$ Torsion structure $$\Z/{2}\Z$$

# Related objects

Show commands: Magma / Oscar / PariGP / SageMath

## Simplified equation

 $$y^2=x^3+9x$$ y^2=x^3+9x (homogenize, simplify) $$y^2z=x^3+9xz^2$$ y^2z=x^3+9xz^2 (dehomogenize, simplify) $$y^2=x^3+9x$$ y^2=x^3+9x (homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, 0, 0, 9, 0])

gp: E = ellinit([0, 0, 0, 9, 0])

magma: E := EllipticCurve([0, 0, 0, 9, 0]);

oscar: E = EllipticCurve([0, 0, 0, 9, 0])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

## Mordell-Weil group structure

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

magma: MordellWeilGroup(E);

### Infinite order Mordell-Weil generator and height

 $P$ = $$\left(4, 10\right)$$ (4, 10) $\hat{h}(P)$ ≈ $1.7772517496792384795883746700$

sage: E.gens()

magma: Generators(E);

gp: E.gen

## Torsion generators

$$\left(0, 0\right)$$

comment: Torsion subgroup

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

oscar: torsion_structure(E)

## Integral points

$$\left(0, 0\right)$$, $$(4,\pm 10)$$

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

## Invariants

 Conductor: $$576$$ = $2^{6} \cdot 3^{2}$ comment: Conductor  sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E);  oscar: conductor(E) Discriminant: $-46656$ = $-1 \cdot 2^{6} \cdot 3^{6}$ comment: Discriminant  sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E);  oscar: discriminant(E) j-invariant: $$1728$$ = $2^{6} \cdot 3^{3}$ comment: j-invariant  sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E);  oscar: j_invariant(E) Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z[\sqrt{-1}]$$ (potential complex multiplication) sage: E.has_cm()  magma: HasComplexMultiplication(E); Sato-Tate group: $N(\mathrm{U}(1))$ Faltings height: $-0.41465319129748201784603640412\dots$ gp: ellheight(E)  magma: FaltingsHeight(E);  oscar: faltings_height(E) Stable Faltings height: $-1.3105329259115095182522750833\dots$ magma: StableFaltingsHeight(E);  oscar: stable_faltings_height(E)

## BSD invariants

 Analytic rank: $1$ sage: E.analytic_rank()  gp: ellanalyticrank(E)  magma: AnalyticRank(E); Regulator: $1.7772517496792384795883746700\dots$ comment: Regulator  sage: E.regulator()  G = E.gen \\ if available matdet(ellheightmatrix(E,G))  magma: Regulator(E); Real period: $2.1409010280752311986343110517\dots$ 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: $L'(E,1)$ ≈ $1.9024600490183925553056020004$ 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

$\displaystyle 1.902460049 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 2.140901 \cdot 1.777252 \cdot 2}{2^2} \approx 1.902460049$

# 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

$$q - 2 q^{5} - 6 q^{13} - 2 q^{17} + O(q^{20})$$

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: 32
comment: Modular degree

sage: E.modular_degree()

gp: ellmoddegree(E)

magma: ModularDegree(E);

$\Gamma_0(N)$-optimal: yes
Manin constant: 1
comment: Manin constant

magma: ManinConstant(E);

## Local data

This elliptic curve is not semistable. There are 2 primes of bad reduction:

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

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 16.192.9.178

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)];

## Isogenies

gp: ellisomat(E)

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2 and 4.
Its isogeny class 576.c consists of 4 curves linked by isogenies of degrees dividing 4.

## Twists

The minimal quadratic twist of this elliptic curve is 32.a4, its twist by $24$.

The minimal quartic twist of this elliptic curve is 32.a3, its quartic twist by $-9$.

## 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 $K$ $2$ $$\Q(\sqrt{-1})$$ $$\Z/2\Z \oplus \Z/2\Z$$ 2.0.4.1-20736.1-CMb1 $2$ $$\Q(\sqrt{6})$$ $$\Z/4\Z$$ 2.2.24.1-32.1-b2 $2$ $$\Q(\sqrt{-6})$$ $$\Z/4\Z$$ Not in database $4$ $$\Q(i, \sqrt{6})$$ $$\Z/2\Z \oplus \Z/4\Z$$ Not in database $8$ $$\Q(\zeta_{24})$$ $$\Z/4\Z \oplus \Z/4\Z$$ Not in database $8$ 8.4.339738624.2 $$\Z/8\Z$$ Not in database $8$ 8.0.84934656.2 $$\Z/8\Z$$ Not in database $8$ 8.2.573308928.1 $$\Z/6\Z$$ Not in database $8$ 8.0.2654208000.1 $$\Z/2\Z \oplus \Z/10\Z$$ Not in database $16$ 16.0.115422332637413376.2 $$\Z/4\Z \oplus \Z/8\Z$$ Not in database $16$ 16.0.328683126924509184.1 $$\Z/6\Z \oplus \Z/6\Z$$ Not in database $16$ deg 16 $$\Z/10\Z$$ Not in database $16$ 16.4.5258930030792146944.1 $$\Z/12\Z$$ Not in database $16$ deg 16 $$\Z/2\Z \oplus \Z/20\Z$$ Not in database $16$ 16.0.5258930030792146944.3 $$\Z/12\Z$$ 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) $\mu$-invariant(s) 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 add add ord ss ss ord ord ss ss ord ss ord ord ss ss - - 3 1,1 1,3 1 1 1,3 1,1 1 1,1 1 1 1,1 1,1 - - 0 0,0 0,0 0 0 0,0 0,0 0 0,0 0 0 0,0 0,0

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

## $p$-adic regulators

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.