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Properties

Label 162.d2
Conductor $162$
Discriminant $-5184$
j-invariant \( \frac{109503}{64} \)
CM no
Rank $0$
Torsion structure \(\Z/{3}\Z\)

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy+y=x^3-x^2+4x-1\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz+yz^2=x^3-x^2z+4xz^2-z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3+69x+22\) Copy content Toggle raw display (homogenize, minimize)

sage: E = EllipticCurve([1, -1, 1, 4, -1])
 
gp: E = ellinit([1, -1, 1, 4, -1])
 
magma: E := EllipticCurve([1, -1, 1, 4, -1]);
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 

Mordell-Weil group structure

\(\Z/{3}\Z\)

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(1, 1\right) \) Copy content Toggle raw display

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\( \left(1, 1\right) \), \( \left(1, -3\right) \) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 162 \)  =  $2 \cdot 3^{4}$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-5184 $  =  $-1 \cdot 2^{6} \cdot 3^{4} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{109503}{64} \)  =  $2^{-6} \cdot 3^{2} \cdot 23^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $-0.59780648594258934269379073217\dots$
Stable Faltings height: $-0.96401058216529257315887247781\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $0$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1$
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);
 
Real period: $2.6051702504152866173475482491\dots$
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
Tamagawa product: $ 6 $  = $ ( 2 \cdot 3 )\cdot1 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $3$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $1$ (exact)
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Special value: $ L(E,1) $ ≈ $ 1.7367801669435244115650321660 $

Modular invariants

Modular form   162.2.a.d

sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 
magma: ModularForm(E);
 

\( q + q^{2} + q^{4} + 3 q^{5} - 4 q^{7} + q^{8} + 3 q^{10} - q^{13} - 4 q^{14} + q^{16} + 3 q^{17} - 4 q^{19} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 12
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1

Local data

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

sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$2$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$3$ $1$ $II$ Additive 1 4 4 0

Galois representations

sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

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
$2$ 2G 4.16.0.2
$3$ 3B.1.1 3.8.0.1
sage: gens = [[9, 4, 8, 5], [7, 3, 0, 1], [9, 4, 4, 9], [1, 9, 9, 10]]
 
sage: GL(2,Integers(12)).subgroup(gens)
 
magma: Gens := [[9, 4, 8, 5], [7, 3, 0, 1], [9, 4, 4, 9], [1, 9, 9, 10]];
 
magma: sub<GL(2,Integers(12))|Gens>;
 

The image of the adelic Galois representation has level $12$, index $128$, genus $1$, and generators

$\left(\begin{array}{rr} 9 & 4 \\ 8 & 5 \end{array}\right),\left(\begin{array}{rr} 7 & 3 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 4 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 10 \end{array}\right)$

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
 

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

Iwasawa invariants

$p$ 2 3
Reduction type split add
$\lambda$-invariant(s) 4 -
$\mu$-invariant(s) 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.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 162.d consists of 2 curves linked by isogenies of degree 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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$3$ 3.1.324.1 \(\Z/12\Z\) Not in database
$6$ 6.0.419904.2 \(\Z/4\Z \oplus \Z/12\Z\) Not in database
$6$ 6.0.177147.2 \(\Z/3\Z \oplus \Z/3\Z\) Not in database
$9$ 9.3.74384733888.1 \(\Z/9\Z\) Not in database
$12$ 12.2.722204136308736.4 \(\Z/24\Z\) Not in database
$18$ 18.0.16599265906765726789632.5 \(\Z/3\Z \oplus \Z/12\Z\) Not in database

We only show fields where the torsion growth is primitive.