↑

Properties

Label 1728e2
Conductor $1728$
Discriminant $-41278242816$
j-invariant \( \frac{9261}{8} \)
CM no
Rank $1$
Torsion structure trivial

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2=x^3+756x+5616\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z=x^3+756xz^2+5616z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3+756x+5616\) Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, 0, 0, 756, 5616])
 
gp: E = ellinit([0, 0, 0, 756, 5616])
 
magma: E := EllipticCurve([0, 0, 0, 756, 5616]);
 
oscar: E = EllipticCurve([0, 0, 0, 756, 5616])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

\(\Z\)

magma: MordellWeilGroup(E);
 

Infinite order Mordell-Weil generator and height

$P$ =  \(\left(-2, 64\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $0.84781944157372172514031143624$

sage: E.gens()
 
magma: Generators(E);
 
gp: E.gen
 

Integral points

\((-2,\pm 64)\), \((40,\pm 316)\) Copy content Toggle raw display

comment: Integral points
 
sage: E.integral_points()
 
magma: IntegralPoints(E);
 

Invariants

Conductor: \( 1728 \)  =  $2^{6} \cdot 3^{3}$
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: $-41278242816 $  =  $-1 \cdot 2^{21} \cdot 3^{9} $
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: \( \frac{9261}{8} \)  =  $2^{-3} \cdot 3^{3} \cdot 7^{3}$
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: $0.72451634274000679710197849719\dots$
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: $-1.1391636446009934355703036127\dots$
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
$abc$ quality: $1.198754152359422\dots$
Szpiro ratio: $4.225205960144686\dots$

BSD invariants

Analytic rank: $1$
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Regulator: $0.84781944157372172514031143624\dots$
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: $0.74413247474456282470323460393\dots$
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: $ 4 $  = $ 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: $1$
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: $1$ ( rounded)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Special value: $ L'(E,1) $ ≈ $ 2.5235599167792273558906286690 $
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

$\displaystyle 2.523559917 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.744132 \cdot 0.847819 \cdot 4}{1^2} \approx 2.523559917$

# 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

Modular form   1728.2.a.c

\( q - 3 q^{5} - q^{7} + 3 q^{11} + 4 q^{13} - 2 q^{19} + O(q^{20}) \) Copy content Toggle raw display

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: 1152
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 2 primes of bad reduction:

prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $4$ $I_{11}^{*}$ Additive 1 6 21 3
$3$ $1$ $IV^{*}$ Additive 1 3 9 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
$3$ 3Cs 9.36.0.1

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, 12, 0, 1], [55, 18, 0, 1], [35, 54, 27, 53], [1, 9, 9, 10], [7, 18, 9, 53], [1, 18, 0, 1], [1, 6, 6, 37], [55, 18, 54, 19], [1, 0, 18, 1]]
 
GL(2,Integers(72)).subgroup(gens)
 
Gens := [[1, 12, 0, 1], [55, 18, 0, 1], [35, 54, 27, 53], [1, 9, 9, 10], [7, 18, 9, 53], [1, 18, 0, 1], [1, 6, 6, 37], [55, 18, 54, 19], [1, 0, 18, 1]];
 
sub<GL(2,Integers(72))|Gens>;
 

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 72 = 2^{3} \cdot 3^{2} \), index $144$, genus $3$, and generators

$\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 55 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 35 & 54 \\ 27 & 53 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 10 \end{array}\right),\left(\begin{array}{rr} 7 & 18 \\ 9 & 53 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 55 & 18 \\ 54 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right)$.

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

The torsion field $K:=\Q(E[72])$ is a degree-$41472$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/72\Z)$.

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 1728e consists of 3 curves linked by isogenies of degrees dividing 9.

Twists

The minimal quadratic twist of this elliptic curve is 54a1, its twist by $8$.

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

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{2}) \) \(\Z/3\Z\) 2.2.8.1-1458.1-k3
$2$ \(\Q(\sqrt{-6}) \) \(\Z/3\Z\) Not in database
$3$ 3.1.216.1 \(\Z/2\Z\) Not in database
$4$ \(\Q(\sqrt{2}, \sqrt{-3})\) \(\Z/3\Z \oplus \Z/3\Z\) Not in database
$6$ 6.0.1119744.1 \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$6$ 6.0.4478976.2 \(\Z/9\Z\) Not in database
$6$ 6.2.373248.1 \(\Z/6\Z\) Not in database
$12$ 12.2.1925877696823296.3 \(\Z/4\Z\) Not in database
$12$ 12.0.1253826625536.1 \(\Z/6\Z \oplus \Z/6\Z\) Not in database
$18$ 18.6.181308248410966445147553792.1 \(\Z/9\Z\) Not in database
$18$ 18.0.7278153735662003552256.1 \(\Z/2\Z \oplus \Z/18\Z\) 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 ss ord ord ord ord ord ord ord ord
$\lambda$-invariant(s) - - 1 1 1 1 1,1 1 1 1 1 1 1 1 1
$\mu$-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

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.