Properties

Label 546f2
Conductor $546$
Discriminant $-2635437714$
j-invariant \( -\frac{5486773802537974663600129}{2635437714} \)
CM no
Rank $0$
Torsion structure trivial

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -3674496, -2711401518])
 
gp: E = ellinit([1, 0, 0, -3674496, -2711401518])
 
magma: E := EllipticCurve([1, 0, 0, -3674496, -2711401518]);
 

\(y^2+xy=x^3-3674496x-2711401518\)  Toggle raw display

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: $-2635437714 $  =  $-1 \cdot 2 \cdot 3 \cdot 7 \cdot 13^{7} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{5486773802537974663600129}{2635437714} \)  =  $-1 \cdot 2^{-1} \cdot 3^{-1} \cdot 7^{-1} \cdot 13^{-7} \cdot 41^{3} \cdot 43^{3} \cdot 100043^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.0465476740267164251249938637\dots$
Stable Faltings height: $2.0465476740267164251249938637\dots$

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: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.054521791001983499995758462845\dots$
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: $49$ = $7^2$ (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) $ ≈ $ 2.6715677590971914997921646794030442560 $

Modular invariants

Modular form   546.2.a.f

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 8232
$ \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$(j)_{-}$
$2$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$3$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$7$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$13$ $1$ $I_{7}$ Non-split multiplicative 1 1 7 7

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
$7$ 7B.1.3 7.48.0.5

$p$-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 5 7 13
Reduction type split split ordinary split nonsplit
$\lambda$-invariant(s) 2 1 0 5 0
$\mu$-invariant(s) 0 0 0 1 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 7.
Its isogeny class 546f consists of 2 curves linked by isogenies of degree 7.

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
$3$ 3.1.2184.1 \(\Z/2\Z\) Not in database
$6$ 6.0.10417365504.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
$6$ \(\Q(\zeta_{7})\) \(\Z/7\Z\) Not in database
$7$ 7.1.4520453669548992.29 \(\Z/7\Z\) Not in database
$8$ 8.2.194365577860272.8 \(\Z/3\Z\) Not in database
$12$ Deg 12 \(\Z/4\Z\) Not in database
$18$ 18.0.4379234125237628624076275712.1 \(\Z/14\Z\) Not in database
$21$ 21.1.3988584188566184927712460097535808514732885044093734132645888.1 \(\Z/14\Z\) 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.