Properties

Label 28.0.282...776.1
Degree $28$
Signature $[0, 14]$
Discriminant $2.826\times 10^{46}$
Root discriminant $45.60$
Ramified primes $2, 29$
Class number $1344$ (GRH)
Class group $[4, 4, 84]$ (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 + 27*x^26 + 325*x^24 + 2300*x^22 + 10626*x^20 + 33649*x^18 + 74613*x^16 + 116280*x^14 + 125970*x^12 + 92378*x^10 + 43758*x^8 + 12376*x^6 + 1820*x^4 + 105*x^2 + 1)
 
gp: K = bnfinit(x^28 + 27*x^26 + 325*x^24 + 2300*x^22 + 10626*x^20 + 33649*x^18 + 74613*x^16 + 116280*x^14 + 125970*x^12 + 92378*x^10 + 43758*x^8 + 12376*x^6 + 1820*x^4 + 105*x^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 105, 0, 1820, 0, 12376, 0, 43758, 0, 92378, 0, 125970, 0, 116280, 0, 74613, 0, 33649, 0, 10626, 0, 2300, 0, 325, 0, 27, 0, 1]);
 

\( x^{28} + 27 x^{26} + 325 x^{24} + 2300 x^{22} + 10626 x^{20} + 33649 x^{18} + 74613 x^{16} + 116280 x^{14} + 125970 x^{12} + 92378 x^{10} + 43758 x^{8} + 12376 x^{6} + 1820 x^{4} + 105 x^{2} + 1 \)

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

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(28261019450929335045240957686456444760176459776\)\(\medspace = 2^{28}\cdot 29^{26}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $45.60$
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:  \(116=2^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{116}(1,·)$, $\chi_{116}(67,·)$, $\chi_{116}(5,·)$, $\chi_{116}(7,·)$, $\chi_{116}(9,·)$, $\chi_{116}(13,·)$, $\chi_{116}(109,·)$, $\chi_{116}(115,·)$, $\chi_{116}(81,·)$, $\chi_{116}(83,·)$, $\chi_{116}(23,·)$, $\chi_{116}(25,·)$, $\chi_{116}(91,·)$, $\chi_{116}(93,·)$, $\chi_{116}(107,·)$, $\chi_{116}(33,·)$, $\chi_{116}(35,·)$, $\chi_{116}(65,·)$, $\chi_{116}(103,·)$, $\chi_{116}(71,·)$, $\chi_{116}(45,·)$, $\chi_{116}(111,·)$, $\chi_{116}(49,·)$, $\chi_{116}(51,·)$, $\chi_{116}(53,·)$, $\chi_{116}(57,·)$, $\chi_{116}(59,·)$, $\chi_{116}(63,·)$$\rbrace$
This is a CM field.

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

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

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

Class group and class number

$C_{4}\times C_{4}\times C_{84}$, which has order $1344$ (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:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -a^{27} - 26 a^{25} - 300 a^{23} - 2024 a^{21} - 8855 a^{19} - 26334 a^{17} - 54264 a^{15} - 77520 a^{13} - 75582 a^{11} - 48620 a^{9} - 19448 a^{7} - 4368 a^{5} - 455 a^{3} - 14 a \) (order $4$)
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:  \( 487075979.1876791 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{14}\cdot 487075979.1876791 \cdot 1344}{4\sqrt{28261019450929335045240957686456444760176459776}}\approx 0.145499075792487$ (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

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-29}) \), \(\Q(i, \sqrt{29})\), 7.7.594823321.1, \(\Q(\zeta_{29})^+\), 14.0.5796901408038404767744.1, 14.0.168110140833113738264576.1

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 ${\href{/LocalNumberField/3.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
$29$29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$
29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$