Properties

Label 28.2.119...424.1
Degree $28$
Signature $[2, 13]$
Discriminant $-1.190\times 10^{47}$
Root discriminant $48.00$
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{28}$ (as 28T1854)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 4*x - 1)
 
gp: K = bnfinit(x^28 - 4*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
 

\( x^{28} - 4 x - 1 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-119031391573998606077977177245018890036706279424\)\(\medspace = -\,2^{30}\cdot 313\cdot 354174511376228618158774900446574177\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $48.00$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 313, 354174511376228618158774900446574177$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{11}$, $\frac{1}{2} a^{26} - \frac{1}{2} a^{12}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{13}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 2934730265132.1826 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{13}\cdot 2934730265132.1826 \cdot 1}{2\sqrt{119031391573998606077977177245018890036706279424}}\approx 0.404674389054564$ (assuming GRH)

Galois group

$S_{28}$ (as 28T1854):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 304888344611713860501504000000
The 3718 conjugacy class representatives for $S_{28}$ are not computed
Character table for $S_{28}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/LocalNumberField/3.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ $19{,}\,{\href{/LocalNumberField/5.9.0.1}{9} }$ $16{,}\,{\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.13.0.1}{13} }{,}\,{\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ $16{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ $22{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $26{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $22{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ $17{,}\,{\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ $19{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $26{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $15{,}\,{\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $25{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $19{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ $19{,}\,{\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.4$x^{6} + x^{2} + 1$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.12.12.22$x^{12} - 52 x^{10} - 7 x^{8} + 32 x^{6} + 35 x^{4} - 44 x^{2} - 29$$2$$6$$12$12T134$[2, 2, 2, 2, 2, 2]^{6}$
313Data not computed
354174511376228618158774900446574177Data not computed