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Properties

Label 14.0.91022458951...6875.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,5^{7}\cdot 71^{13}$
Root discriminant $117.09$
Ramified primes $5, 71$
Class number $7424$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 116]$ (GRH)
Galois group $C_{14}$ (as 14T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![867961, 913390, 2049631, 804688, 790947, 274929, 152364, 14729, 14518, 937, 1819, -82, 74, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 74*x^12 - 82*x^11 + 1819*x^10 + 937*x^9 + 14518*x^8 + 14729*x^7 + 152364*x^6 + 274929*x^5 + 790947*x^4 + 804688*x^3 + 2049631*x^2 + 913390*x + 867961)
 
gp: K = bnfinit(x^14 - x^13 + 74*x^12 - 82*x^11 + 1819*x^10 + 937*x^9 + 14518*x^8 + 14729*x^7 + 152364*x^6 + 274929*x^5 + 790947*x^4 + 804688*x^3 + 2049631*x^2 + 913390*x + 867961, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 74 x^{12} - 82 x^{11} + 1819 x^{10} + 937 x^{9} + 14518 x^{8} + 14729 x^{7} + 152364 x^{6} + 274929 x^{5} + 790947 x^{4} + 804688 x^{3} + 2049631 x^{2} + 913390 x + 867961 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-91022458951991999260243046875=-\,5^{7}\cdot 71^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $117.09$
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:  $5, 71$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(355=5\cdot 71\)
Dirichlet character group:    $\lbrace$$\chi_{355}(1,·)$, $\chi_{355}(34,·)$, $\chi_{355}(101,·)$, $\chi_{355}(39,·)$, $\chi_{355}(264,·)$, $\chi_{355}(354,·)$, $\chi_{355}(321,·)$, $\chi_{355}(239,·)$, $\chi_{355}(116,·)$, $\chi_{355}(94,·)$, $\chi_{355}(91,·)$, $\chi_{355}(316,·)$, $\chi_{355}(254,·)$, $\chi_{355}(261,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{4} + \frac{1}{5}$, $\frac{1}{85} a^{9} + \frac{1}{17} a^{8} - \frac{7}{17} a^{7} + \frac{23}{85} a^{5} + \frac{1}{17} a^{4} + \frac{8}{17} a^{3} + \frac{1}{17} a^{2} - \frac{19}{85} a - \frac{1}{17}$, $\frac{1}{85} a^{10} + \frac{8}{85} a^{8} + \frac{1}{17} a^{7} + \frac{23}{85} a^{6} - \frac{5}{17} a^{5} - \frac{36}{85} a^{4} - \frac{5}{17} a^{3} + \frac{41}{85} a^{2} + \frac{1}{17} a + \frac{8}{85}$, $\frac{1}{85} a^{11} - \frac{1}{85} a^{8} - \frac{37}{85} a^{7} - \frac{5}{17} a^{6} + \frac{7}{17} a^{5} + \frac{37}{85} a^{4} - \frac{24}{85} a^{3} - \frac{7}{17} a^{2} - \frac{2}{17} a - \frac{11}{85}$, $\frac{1}{425} a^{12} - \frac{2}{425} a^{11} + \frac{2}{425} a^{10} - \frac{1}{425} a^{9} + \frac{32}{425} a^{8} + \frac{144}{425} a^{7} + \frac{131}{425} a^{6} + \frac{172}{425} a^{5} + \frac{9}{25} a^{4} - \frac{207}{425} a^{3} + \frac{57}{425} a^{2} - \frac{66}{425} a + \frac{89}{425}$, $\frac{1}{2790258039611270274423936857318375} a^{13} + \frac{1788682442382558177704542213931}{2790258039611270274423936857318375} a^{12} - \frac{16098170771303663424352464930229}{2790258039611270274423936857318375} a^{11} + \frac{513885025136235302560470645046}{558051607922254054884787371463675} a^{10} - \frac{1387851611337982946832443332636}{2790258039611270274423936857318375} a^{9} + \frac{18733919790606881746602490863676}{558051607922254054884787371463675} a^{8} - \frac{865813515791865295977899084138662}{2790258039611270274423936857318375} a^{7} - \frac{13914018633774920607983047486482}{558051607922254054884787371463675} a^{6} + \frac{1379475617265813908785354265450424}{2790258039611270274423936857318375} a^{5} + \frac{347644256330063943951289911047557}{2790258039611270274423936857318375} a^{4} + \frac{1172548131047041409663975744521961}{2790258039611270274423936857318375} a^{3} + \frac{132316879430791522745993097392011}{558051607922254054884787371463675} a^{2} + \frac{283900232358676188659993419521351}{2790258039611270274423936857318375} a + \frac{356013750557053694174746451003892}{2790258039611270274423936857318375}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{116}$, which has order $7424$ (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:  $6$
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:  \( 315114.6966253571 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{14}$ (as 14T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{-355}) \), 7.7.128100283921.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ R ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$71$71.14.13.11$x^{14} + 9088$$14$$1$$13$$C_{14}$$[\ ]_{14}$