LMFDB - Elliptic curve with Cremona label 1728r1 (LMFDB label 1728.h1)

Show commands: Magma / Oscar / PariGP / SageMath

Simplified equation

$y^2=x^3-6x-6$ y^2=x^3-6x-6	(homogenize, simplify)
$y^2z=x^3-6xz^2-6z^3$ y^2z=x^3-6xz^2-6z^3	(dehomogenize, simplify)
$y^2=x^3-6x-6$ y^2=x^3-6x-6	(homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([0, 0, 0, -6, -6])

gp: E = ellinit([0, 0, 0, -6, -6])

magma: E := EllipticCurve([0, 0, 0, -6, -6]);

oscar: E = elliptic_curve([0, 0, 0, -6, -6])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

Invariants

Conductor:	$N$	=	$1728$	=	$2^{6} \cdot 3^{3}$	comment: Conductor sage: E.conductor().factor() gp: ellglobalred(E)[1] magma: Conductor(E); oscar: conductor(E)
Discriminant:	$\Delta$	=	$-1728$	=	$-1 \cdot 2^{6} \cdot 3^{3}$	comment: Discriminant sage: E.discriminant().factor() gp: E.disc magma: Discriminant(E); oscar: discriminant(E)
j-invariant:	$j$	=	$-13824$	=	$-1 \cdot 2^{9} \cdot 3^{3}$	comment: j-invariant sage: E.j_invariant().factor() gp: E.j magma: jInvariant(E); oscar: j_invariant(E)
Endomorphism ring:	$\mathrm{End}(E)$	=	$\Z$
Geometric endomorphism ring:	$\mathrm{End}(E_{\overline{\Q}})$	=	$\Z$ (no potential complex multiplication)			sage: E.has_cm() magma: HasComplexMultiplication(E);
Sato-Tate group:	$\mathrm{ST}(E)$	=	$\mathrm{SU}(2)$
Faltings height:	$h_{\mathrm{Faltings}}$	≈	$-0.62917836822511953336480818753$			gp: ellheight(E) magma: FaltingsHeight(E); oscar: faltings_height(E)
Stable Faltings height:	$h_{\mathrm{stable}}$	≈	$-1.2504050306721196109222355575$			magma: StableFaltingsHeight(E); oscar: stable_faltings_height(E)
$abc$ quality:	$Q$	≈	$1.2262943855309167$
Szpiro ratio:	$\sigma_{m}$	≈	$2.2947427391318267$

BSD invariants

Analytic rank:	$r_{\mathrm{an}}$	=	$0$	sage: E.analytic_rank() gp: ellanalyticrank(E) magma: AnalyticRank(E);
Mordell-Weil rank:	$r$	=	$0$	comment: Rank sage: E.rank() gp: [lower,upper] = ellrank(E) magma: Rank(E);
Regulator:	$\mathrm{Reg}(E/\Q)$	=	$1$	comment: Regulator sage: E.regulator() G = E.gen \\ if available matdet(ellheightmatrix(E,G)) magma: Regulator(E);
Real period:	$\Omega$	≈	$1.5188944813290509624956164484$	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:	$\prod_{p}c_p$	=	$1$	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:	$\#E(\Q)_{\mathrm{tor}}$	=	$1$	comment: Torsion order sage: E.torsion_order() gp: elltors(E)[1] magma: Order(TorsionSubgroup(E)); oscar: prod(torsion_structure(E)[1])
Special value:	$L(E,1)$	≈	$1.5188944813290509624956164483$	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);
Analytic order of Ш:	Ш ${}_{\mathrm{an}}$	=	$1$ (exact)	comment: Order of Sha sage: E.sha().an_numerical() magma: MordellWeilShaInformation(E);

$\displaystyle 1.518894481 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.518894 \cdot 1.000000 \cdot 1}{1^2} \approx 1.518894481$

# 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 analytic 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

Modular form 1728.2.a.h

q - 2 q^{5} + 3 q^{7} - 6 q^{11} + 3 q^{13} - 2 q^{17} + 3 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);

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

Modular degree:	96	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 at primes of bad reduction

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

$p$	Tamagawa number	Kodaira symbol	Reduction type	Root number	$\mathrm{ord}_p(N)$	$\mathrm{ord}_p(\Delta)$	$\mathrm{ord}_p(\mathrm{den}(j))$
$2$	$1$	$II$	additive	-1	6	6	0
$3$	$1$	$II$	additive	1	3	3	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
$3$	3Nn	3.3.0.1

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 = [[7, 0, 6, 7], [13, 12, 12, 13], [10, 5, 7, 10], [3, 4, 8, 3], [4, 9, 3, 7], [11, 0, 0, 23], [1, 0, 12, 1], [5, 0, 0, 5], [1, 12, 0, 1], [23, 18, 18, 23], [19, 19, 13, 14]]

GL(2,Integers(24)).subgroup(gens)

Gens := [[7, 0, 6, 7], [13, 12, 12, 13], [10, 5, 7, 10], [3, 4, 8, 3], [4, 9, 3, 7], [11, 0, 0, 23], [1, 0, 12, 1], [5, 0, 0, 5], [1, 12, 0, 1], [23, 18, 18, 23], [19, 19, 13, 14]];

sub<GL(2,Integers(24))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 24.24.1-12.g.1.1, level $24 = 2^{3} \cdot 3$ , index $24$ , genus $1$ , and generators

$\left(\begin{array}{rr} 7 & 0 \\ 6 & 7 \end{array}\right),\left(\begin{array}{rr} 13 & 12 \\ 12 & 13 \end{array}\right),\left(\begin{array}{rr} 10 & 5 \\ 7 & 10 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 3 \end{array}\right),\left(\begin{array}{rr} 4 & 9 \\ 3 & 7 \end{array}\right),\left(\begin{array}{rr} 11 & 0 \\ 0 & 23 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 18 & 23 \end{array}\right),\left(\begin{array}{rr} 19 & 19 \\ 13 & 14 \end{array}\right)$ .

The torsion field $K:=\Q(E[24])$ is a degree- $3072$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24\Z)$ .

The table below list all primes $\ell$ for which the Serre invariants associated to the mod- $\ell$ Galois representation are exceptional.

$\ell$	Reduction type	Serre weight	Serre conductor
$2$	additive	$2$	$27 = 3^{3}$
$3$	additive	$6$	$64 = 2^{6}$

Isogenies

gp: ellisomat(E)

This curve has no rational isogenies. Its isogeny class 1728r consists of this curve only.

Twists

The minimal quadratic twist of this elliptic curve is 864i1, its twist by $24$ .

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$	$K$	$E(K)_{\rm tors}$	Base change curve
$3$	3.1.108.1	$\Z/2\Z$	not in database
$6$	6.0.34992.1	$\Z/2\Z \oplus \Z/2\Z$	not in database
$8$	8.2.573308928.1	$\Z/3\Z$	not in database
$12$	12.2.3851755393646592.8	$\Z/4\Z$	not in database
$16$	16.0.328683126924509184.1	$\Z/3\Z \oplus \Z/3\Z$	not in database

We only show fields where the torsion growth is primitive.

Iwasawa invariants

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47
Reduction type	add	add	ord	ord	ord	ord	ord	ord	ord	ord	ss	ord	ord	ord	ord
$\lambda$ -invariant(s)	-	-	0	0	0	0	0	0	0	0	0,0	0	0	0	0
$\mu$ -invariant(s)	-	-	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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

Mordell-Weil group structure

Invariants

BSD invariants

BSD formula

Modular invariants

Local data at primes of bad reduction

Galois representations

Isogenies

Twists

Growth of torsion in number fields

Iwasawa invariants

$p$ -adic regulators

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1728r1

Conductor	$1728$
Discriminant	$-1728$
j-invariant	$-13824$
CM	no
Rank	$0$
Torsion structure	trivial

Properties

Related objects

Downloads

Learn more

Mordell-Weil group structure

Invariants

Local data at primes of bad reduction

Growth of torsion in number fields

ppp-adic regulators

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

$p$ -adic regulators