Properties

Label 546.d1
Conductor $546$
Discriminant $-36641439744$
j-invariant \( -\frac{1956469094246217097}{36641439744} \)
CM no
Rank $0$
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+xy+y=x^3-26057x-1621108\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+xyz+yz^2=x^3-26057xz^2-1621108z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-33769251x-75533095458\) Copy content Toggle raw display (homogenize, minimize)

sage: E = EllipticCurve([1, 0, 1, -26057, -1621108])
 
gp: E = ellinit([1, 0, 1, -26057, -1621108])
 
magma: E := EllipticCurve([1, 0, 1, -26057, -1621108]);
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 

Mordell-Weil group structure

trivial

Integral points

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

None

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 546 \)  =  $2 \cdot 3 \cdot 7 \cdot 13$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-36641439744 $  =  $-1 \cdot 2^{27} \cdot 3 \cdot 7 \cdot 13 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{1956469094246217097}{36641439744} \)  =  $-1 \cdot 2^{-27} \cdot 3^{-1} \cdot 7^{-1} \cdot 13^{-1} \cdot 19^{3} \cdot 65827^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.1514593355799436739008714851\dots$
Stable Faltings height: $1.1514593355799436739008714851\dots$

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: $0.18788407213259006409145542425\dots$
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: $ 1 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $1$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $9$ = $3^2$ (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) $ ≈ $ 1.6909566491933105768230988183 $

Modular invariants

Modular form   546.2.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);
 

\( q - q^{2} + q^{3} + q^{4} + 3 q^{5} - q^{6} + q^{7} - q^{8} + q^{9} - 3 q^{10} + 3 q^{11} + q^{12} + q^{13} - q^{14} + 3 q^{15} + q^{16} - 3 q^{17} - q^{18} - 7 q^{19} + O(q^{20}) \) Copy content Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 1944
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is semistable. There are 4 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$(j)_{-}$
$2$ $1$ $I_{27}$ Non-split multiplicative 1 1 27 27
$3$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$7$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$13$ $1$ $I_{1}$ Split multiplicative -1 1 1 1

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
$3$ 3B.1.2 9.24.0.3
sage: gens = [[1, 18, 0, 1], [4915, 3294, 4923, 1073], [3277, 18, 3285, 163], [1, 18, 10, 181], [10, 9, 81, 73], [1639, 18, 1647, 163], [7, 18, 1737, 6205], [1, 0, 18, 1], [6535, 18, 6534, 19], [13, 18, 3897, 511]]
 
sage: GL(2,Integers(6552)).subgroup(gens)
 
magma: Gens := [[1, 18, 0, 1], [4915, 3294, 4923, 1073], [3277, 18, 3285, 163], [1, 18, 10, 181], [10, 9, 81, 73], [1639, 18, 1647, 163], [7, 18, 1737, 6205], [1, 0, 18, 1], [6535, 18, 6534, 19], [13, 18, 3897, 511]];
 
magma: sub<GL(2,Integers(6552))|Gens>;
 

The image of the adelic Galois representation has level $6552$, index $144$, genus $3$, and generators

$\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4915 & 3294 \\ 4923 & 1073 \end{array}\right),\left(\begin{array}{rr} 3277 & 18 \\ 3285 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 1639 & 18 \\ 1647 & 163 \end{array}\right),\left(\begin{array}{rr} 7 & 18 \\ 1737 & 6205 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 6535 & 18 \\ 6534 & 19 \end{array}\right),\left(\begin{array}{rr} 13 & 18 \\ 3897 & 511 \end{array}\right)$

$p$-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 7 13
Reduction type nonsplit split split split
$\lambda$-invariant(s) 2 9 1 1
$\mu$-invariant(s) 0 2 0 0

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

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3 and 9.
Its isogeny class 546.d consists of 3 curves linked by isogenies of degrees dividing 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-3}) \) \(\Z/3\Z\) 2.0.3.1-99372.5-k1
$3$ 3.1.2184.1 \(\Z/2\Z\) Not in database
$3$ 3.1.2012283.4 \(\Z/3\Z\) Not in database
$6$ 6.0.10417365504.1 \(\Z/2\Z \oplus \Z/2\Z\) Not in database
$6$ 6.0.12147848616267.2 \(\Z/3\Z \oplus \Z/3\Z\) Not in database
$6$ 6.0.1349760957363.10 \(\Z/9\Z\) Not in database
$6$ 6.0.1466954307.3 \(\Z/9\Z\) Not in database
$6$ 6.0.14309568.3 \(\Z/6\Z\) Not in database
$9$ 9.1.1138937212106225811482112.1 \(\Z/6\Z\) Not in database
$12$ deg 12 \(\Z/4\Z\) Not in database
$12$ deg 12 \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$18$ 18.0.1306849698240310085247366316592533934392827.8 \(\Z/3\Z \oplus \Z/9\Z\) Not in database
$18$ 18.0.3891533919360906009465097143985563592854287941632.2 \(\Z/3\Z \oplus \Z/6\Z\) Not in database
$18$ 18.0.5338180959342806597345812268841651019004510208.1 \(\Z/18\Z\) Not in database
$18$ 18.0.56748612942862716458053795232717136986112.1 \(\Z/18\Z\) Not in database
$18$ 18.0.181314348370863332792997806132575378918266983776518144.1 \(\Z/2\Z \oplus \Z/6\Z\) 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.