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Properties

Label 28.28.1342782773...9536.1
Degree $28$
Signature $[28, 0]$
Discriminant $2^{42}\cdot 29^{27}$
Root discriminant $72.73$
Ramified primes $2, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{28}$ (as 28T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![475136, 0, -8314880, 0, 43237376, 0, -105005056, 0, 144381952, 0, -124693504, 0, 71938560, 0, -28775424, 0, 8145984, 0, -1643488, 0, 234784, 0, -23200, 0, 1508, 0, -58, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 58*x^26 + 1508*x^24 - 23200*x^22 + 234784*x^20 - 1643488*x^18 + 8145984*x^16 - 28775424*x^14 + 71938560*x^12 - 124693504*x^10 + 144381952*x^8 - 105005056*x^6 + 43237376*x^4 - 8314880*x^2 + 475136)
 
gp: K = bnfinit(x^28 - 58*x^26 + 1508*x^24 - 23200*x^22 + 234784*x^20 - 1643488*x^18 + 8145984*x^16 - 28775424*x^14 + 71938560*x^12 - 124693504*x^10 + 144381952*x^8 - 105005056*x^6 + 43237376*x^4 - 8314880*x^2 + 475136, 1)
 

Normalized defining polynomial

\( x^{28} - 58 x^{26} + 1508 x^{24} - 23200 x^{22} + 234784 x^{20} - 1643488 x^{18} + 8145984 x^{16} - 28775424 x^{14} + 71938560 x^{12} - 124693504 x^{10} + 144381952 x^{8} - 105005056 x^{6} + 43237376 x^{4} - 8314880 x^{2} + 475136 \)

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

Invariants

Degree:  $28$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[28, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13427827737836760536055607671312169337571202392129536=2^{42}\cdot 29^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.73$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(232=2^{3}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{232}(1,·)$, $\chi_{232}(3,·)$, $\chi_{232}(147,·)$, $\chi_{232}(81,·)$, $\chi_{232}(129,·)$, $\chi_{232}(9,·)$, $\chi_{232}(11,·)$, $\chi_{232}(161,·)$, $\chi_{232}(43,·)$, $\chi_{232}(75,·)$, $\chi_{232}(19,·)$, $\chi_{232}(121,·)$, $\chi_{232}(225,·)$, $\chi_{232}(25,·)$, $\chi_{232}(155,·)$, $\chi_{232}(33,·)$, $\chi_{232}(163,·)$, $\chi_{232}(131,·)$, $\chi_{232}(65,·)$, $\chi_{232}(209,·)$, $\chi_{232}(169,·)$, $\chi_{232}(171,·)$, $\chi_{232}(27,·)$, $\chi_{232}(49,·)$, $\chi_{232}(211,·)$, $\chi_{232}(99,·)$, $\chi_{232}(57,·)$, $\chi_{232}(195,·)$$\rbrace$
This is not a CM field.

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

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

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

Class group and class number

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

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $27$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 170028857236944540 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{28}$ (as 28T1):

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

Intermediate fields

\(\Q(\sqrt{29}) \), 4.4.1560896.1, 7.7.594823321.1, \(\Q(\zeta_{29})^+\)

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 $28$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ $28$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ $28$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R $28$ $28$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ $28$ $28$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{28}$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
29Data not computed