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Properties

Label 28.28.5002255640...0000.1
Degree $28$
Signature $[28, 0]$
Discriminant $2^{28}\cdot 5^{14}\cdot 29^{27}$
Root discriminant $115.00$
Ramified primes $2, 5, 29$
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![177001953125, 0, -1239013671875, 0, 2577148437500, 0, -2503515625000, 0, 1376933593750, 0, -475667968750, 0, 109769531250, 0, -17563125000, 0, 1988765625, 0, -160496875, 0, 9171250, 0, -362500, 0, 9425, 0, -145, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 145*x^26 + 9425*x^24 - 362500*x^22 + 9171250*x^20 - 160496875*x^18 + 1988765625*x^16 - 17563125000*x^14 + 109769531250*x^12 - 475667968750*x^10 + 1376933593750*x^8 - 2503515625000*x^6 + 2577148437500*x^4 - 1239013671875*x^2 + 177001953125)
 
gp: K = bnfinit(x^28 - 145*x^26 + 9425*x^24 - 362500*x^22 + 9171250*x^20 - 160496875*x^18 + 1988765625*x^16 - 17563125000*x^14 + 109769531250*x^12 - 475667968750*x^10 + 1376933593750*x^8 - 2503515625000*x^6 + 2577148437500*x^4 - 1239013671875*x^2 + 177001953125, 1)
 

Normalized defining polynomial

\( x^{28} - 145 x^{26} + 9425 x^{24} - 362500 x^{22} + 9171250 x^{20} - 160496875 x^{18} + 1988765625 x^{16} - 17563125000 x^{14} + 109769531250 x^{12} - 475667968750 x^{10} + 1376933593750 x^{8} - 2503515625000 x^{6} + 2577148437500 x^{4} - 1239013671875 x^{2} + 177001953125 \)

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:  \(5002255640118107399365159746748272082794905600000000000000=2^{28}\cdot 5^{14}\cdot 29^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $115.00$
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:  $2, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(580=2^{2}\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{580}(1,·)$, $\chi_{580}(259,·)$, $\chi_{580}(81,·)$, $\chi_{580}(519,·)$, $\chi_{580}(521,·)$, $\chi_{580}(141,·)$, $\chi_{580}(79,·)$, $\chi_{580}(401,·)$, $\chi_{580}(19,·)$, $\chi_{580}(121,·)$, $\chi_{580}(341,·)$, $\chi_{580}(279,·)$, $\chi_{580}(281,·)$, $\chi_{580}(539,·)$, $\chi_{580}(159,·)$, $\chi_{580}(161,·)$, $\chi_{580}(99,·)$, $\chi_{580}(39,·)$, $\chi_{580}(379,·)$, $\chi_{580}(361,·)$, $\chi_{580}(359,·)$, $\chi_{580}(559,·)$, $\chi_{580}(241,·)$, $\chi_{580}(181,·)$, $\chi_{580}(119,·)$, $\chi_{580}(441,·)$, $\chi_{580}(479,·)$, $\chi_{580}(381,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{25} a^{4}$, $\frac{1}{25} a^{5}$, $\frac{1}{125} a^{6}$, $\frac{1}{125} a^{7}$, $\frac{1}{625} a^{8}$, $\frac{1}{625} a^{9}$, $\frac{1}{3125} a^{10}$, $\frac{1}{3125} a^{11}$, $\frac{1}{15625} a^{12}$, $\frac{1}{15625} a^{13}$, $\frac{1}{78125} a^{14}$, $\frac{1}{78125} a^{15}$, $\frac{1}{390625} a^{16}$, $\frac{1}{390625} a^{17}$, $\frac{1}{1953125} a^{18}$, $\frac{1}{1953125} a^{19}$, $\frac{1}{9765625} a^{20}$, $\frac{1}{9765625} a^{21}$, $\frac{1}{48828125} a^{22}$, $\frac{1}{48828125} a^{23}$, $\frac{1}{244140625} a^{24}$, $\frac{1}{244140625} a^{25}$, $\frac{1}{1220703125} a^{26}$, $\frac{1}{1220703125} a^{27}$

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

\(\Q(\sqrt{29}) \), 4.4.9755600.2, 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 $28$ R ${\href{/LocalNumberField/7.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ $28$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{4}$ R $28$ $28$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{7}$ $28$ $28$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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

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
$5$5.14.7.2$x^{14} - 15625 x^{2} + 156250$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
5.14.7.2$x^{14} - 15625 x^{2} + 156250$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
29Data not computed