Properties

Label 28.28.463...984.1
Degree $28$
Signature $[28, 0]$
Discriminant $4.630\times 10^{50}$
Root discriminant $64.49$
Ramified primes $2, 29$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 54*x^26 + 1300*x^24 - 18400*x^22 + 170016*x^20 - 1076768*x^18 + 4775232*x^16 - 14883840*x^14 + 32248320*x^12 - 47297536*x^10 + 44808192*x^8 - 25346048*x^6 + 7454720*x^4 - 860160*x^2 + 16384)
 
gp: K = bnfinit(x^28 - 54*x^26 + 1300*x^24 - 18400*x^22 + 170016*x^20 - 1076768*x^18 + 4775232*x^16 - 14883840*x^14 + 32248320*x^12 - 47297536*x^10 + 44808192*x^8 - 25346048*x^6 + 7454720*x^4 - 860160*x^2 + 16384, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16384, 0, -860160, 0, 7454720, 0, -25346048, 0, 44808192, 0, -47297536, 0, 32248320, 0, -14883840, 0, 4775232, 0, -1076768, 0, 170016, 0, -18400, 0, 1300, 0, -54, 0, 1]);
 

\( x^{28} - 54 x^{26} + 1300 x^{24} - 18400 x^{22} + 170016 x^{20} - 1076768 x^{18} + 4775232 x^{16} - 14883840 x^{14} + 32248320 x^{12} - 47297536 x^{10} + 44808192 x^{8} - 25346048 x^{6} + 7454720 x^{4} - 860160 x^{2} + 16384 \)

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

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[28, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(463028542684026225381227850734902390950731116969984\)\(\medspace = 2^{42}\cdot 29^{26}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $64.49$
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:  $2, 29$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $28$
This field is Galois and abelian over $\Q$.
Conductor:  \(232=2^{3}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{232}(129,·)$, $\chi_{232}(5,·)$, $\chi_{232}(1,·)$, $\chi_{232}(9,·)$, $\chi_{232}(109,·)$, $\chi_{232}(13,·)$, $\chi_{232}(45,·)$, $\chi_{232}(141,·)$, $\chi_{232}(209,·)$, $\chi_{232}(149,·)$, $\chi_{232}(121,·)$, $\chi_{232}(25,·)$, $\chi_{232}(93,·)$, $\chi_{232}(197,·)$, $\chi_{232}(161,·)$, $\chi_{232}(165,·)$, $\chi_{232}(65,·)$, $\chi_{232}(81,·)$, $\chi_{232}(169,·)$, $\chi_{232}(225,·)$, $\chi_{232}(173,·)$, $\chi_{232}(49,·)$, $\chi_{232}(181,·)$, $\chi_{232}(57,·)$, $\chi_{232}(33,·)$, $\chi_{232}(117,·)$, $\chi_{232}(125,·)$, $\chi_{232}(53,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{4096} a^{24}$, $\frac{1}{4096} a^{25}$, $\frac{1}{8192} a^{26}$, $\frac{1}{8192} a^{27}$

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:  $27$
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:  \( 18257396114183336 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{28}\cdot(2\pi)^{0}\cdot 18257396114183336 \cdot 1}{2\sqrt{463028542684026225381227850734902390950731116969984}}\approx 0.113879313246372$ (assuming GRH)

Galois group

$C_2\times C_{14}$ (as 28T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
An abelian group of order 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$ is not computed

Intermediate fields

\(\Q(\sqrt{58}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{2}, \sqrt{29})\), 7.7.594823321.1, 14.14.21518098026638558497865728.1, 14.14.742003380228915810271232.1, \(\Q(\zeta_{29})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
2Data not computed
29Data not computed