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Properties

Label 28.28.6422488380...2384.1
Degree $28$
Signature $[28, 0]$
Discriminant $2^{42}\cdot 3^{14}\cdot 29^{27}$
Root discriminant $125.97$
Ramified primes $2, 3, 29$
Class number Not computed
Class group Not computed
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![2272560758784, 0, -13256604426240, 0, 22978114338816, 0, -18601330655232, 0, 8525609883648, 0, -2454342239232, 0, 471988892160, 0, -62931852288, 0, 5938422336, 0, -399367584, 0, 19017504, 0, -626400, 0, 13572, 0, -174, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 174*x^26 + 13572*x^24 - 626400*x^22 + 19017504*x^20 - 399367584*x^18 + 5938422336*x^16 - 62931852288*x^14 + 471988892160*x^12 - 2454342239232*x^10 + 8525609883648*x^8 - 18601330655232*x^6 + 22978114338816*x^4 - 13256604426240*x^2 + 2272560758784)
 
gp: K = bnfinit(x^28 - 174*x^26 + 13572*x^24 - 626400*x^22 + 19017504*x^20 - 399367584*x^18 + 5938422336*x^16 - 62931852288*x^14 + 471988892160*x^12 - 2454342239232*x^10 + 8525609883648*x^8 - 18601330655232*x^6 + 22978114338816*x^4 - 13256604426240*x^2 + 2272560758784, 1)
 

Normalized defining polynomial

\( x^{28} - 174 x^{26} + 13572 x^{24} - 626400 x^{22} + 19017504 x^{20} - 399367584 x^{18} + 5938422336 x^{16} - 62931852288 x^{14} + 471988892160 x^{12} - 2454342239232 x^{10} + 8525609883648 x^{8} - 18601330655232 x^{6} + 22978114338816 x^{4} - 13256604426240 x^{2} + 2272560758784 \)

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:  \(64224883807413352704377353768048295264353596334281414672384=2^{42}\cdot 3^{14}\cdot 29^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $125.97$
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:  $2, 3, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(696=2^{3}\cdot 3\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{696}(1,·)$, $\chi_{696}(361,·)$, $\chi_{696}(389,·)$, $\chi_{696}(673,·)$, $\chi_{696}(265,·)$, $\chi_{696}(241,·)$, $\chi_{696}(269,·)$, $\chi_{696}(77,·)$, $\chi_{696}(461,·)$, $\chi_{696}(529,·)$, $\chi_{696}(533,·)$, $\chi_{696}(121,·)$, $\chi_{696}(25,·)$, $\chi_{696}(101,·)$, $\chi_{696}(221,·)$, $\chi_{696}(485,·)$, $\chi_{696}(653,·)$, $\chi_{696}(289,·)$, $\chi_{696}(677,·)$, $\chi_{696}(49,·)$, $\chi_{696}(169,·)$, $\chi_{696}(365,·)$, $\chi_{696}(625,·)$, $\chi_{696}(437,·)$, $\chi_{696}(457,·)$, $\chi_{696}(313,·)$, $\chi_{696}(293,·)$, $\chi_{696}(317,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{6} a^{2}$, $\frac{1}{6} a^{3}$, $\frac{1}{36} a^{4}$, $\frac{1}{36} a^{5}$, $\frac{1}{216} a^{6}$, $\frac{1}{216} a^{7}$, $\frac{1}{1296} a^{8}$, $\frac{1}{1296} a^{9}$, $\frac{1}{7776} a^{10}$, $\frac{1}{7776} a^{11}$, $\frac{1}{46656} a^{12}$, $\frac{1}{46656} a^{13}$, $\frac{1}{279936} a^{14}$, $\frac{1}{279936} a^{15}$, $\frac{1}{1679616} a^{16}$, $\frac{1}{1679616} a^{17}$, $\frac{1}{10077696} a^{18}$, $\frac{1}{10077696} a^{19}$, $\frac{1}{60466176} a^{20}$, $\frac{1}{60466176} a^{21}$, $\frac{1}{362797056} a^{22}$, $\frac{1}{362797056} a^{23}$, $\frac{1}{2176782336} a^{24}$, $\frac{1}{2176782336} a^{25}$, $\frac{1}{13060694016} a^{26}$, $\frac{1}{13060694016} a^{27}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

\(\Q(\sqrt{29}) \), 4.4.14048064.1, 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 R ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ $28$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R $28$ $28$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ $28$ $28$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{28}$

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
3Data not computed
29Data not computed