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Properties

Label 14.0.34859859155...6431.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,41^{7}\cdot 151^{7}$
Root discriminant $78.68$
Ramified primes $41, 151$
Class number $14$ (GRH)
Class group $[14]$ (GRH)
Galois group $D_{7}$ (as 14T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![23001625, 14635125, 6295270, 434497, -298373, -57007, 25724, -3403, -3339, -1422, 184, -23, 16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 16*x^12 - 23*x^11 + 184*x^10 - 1422*x^9 - 3339*x^8 - 3403*x^7 + 25724*x^6 - 57007*x^5 - 298373*x^4 + 434497*x^3 + 6295270*x^2 + 14635125*x + 23001625)
 
gp: K = bnfinit(x^14 + 16*x^12 - 23*x^11 + 184*x^10 - 1422*x^9 - 3339*x^8 - 3403*x^7 + 25724*x^6 - 57007*x^5 - 298373*x^4 + 434497*x^3 + 6295270*x^2 + 14635125*x + 23001625, 1)
 

Normalized defining polynomial

\( x^{14} + 16 x^{12} - 23 x^{11} + 184 x^{10} - 1422 x^{9} - 3339 x^{8} - 3403 x^{7} + 25724 x^{6} - 57007 x^{5} - 298373 x^{4} + 434497 x^{3} + 6295270 x^{2} + 14635125 x + 23001625 \)

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:  \(-348598591551339822624996431=-\,41^{7}\cdot 151^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.68$
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:  $41, 151$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{25} a^{8} - \frac{2}{25} a^{7} - \frac{2}{25} a^{6} + \frac{1}{25} a^{5} - \frac{2}{25} a^{4} + \frac{3}{25} a^{3} + \frac{1}{25} a^{2}$, $\frac{1}{25} a^{9} - \frac{1}{25} a^{7} + \frac{2}{25} a^{6} - \frac{1}{25} a^{4} + \frac{2}{25} a^{3} - \frac{3}{25} a^{2}$, $\frac{1}{125} a^{10} - \frac{2}{125} a^{9} - \frac{1}{125} a^{8} - \frac{11}{125} a^{7} + \frac{11}{125} a^{6} + \frac{4}{125} a^{5} + \frac{4}{125} a^{4} + \frac{58}{125} a^{3} - \frac{59}{125} a^{2} + \frac{9}{25} a - \frac{2}{5}$, $\frac{1}{625} a^{11} - \frac{2}{625} a^{10} - \frac{1}{625} a^{9} + \frac{9}{625} a^{8} + \frac{21}{625} a^{7} + \frac{39}{625} a^{6} - \frac{1}{625} a^{5} + \frac{18}{625} a^{4} + \frac{201}{625} a^{3} - \frac{52}{125} a^{2} - \frac{6}{25} a + \frac{2}{5}$, $\frac{1}{26778125} a^{12} - \frac{224}{281875} a^{11} + \frac{21}{281875} a^{10} + \frac{77307}{26778125} a^{9} - \frac{42946}{26778125} a^{8} - \frac{2344989}{26778125} a^{7} - \frac{2010403}{26778125} a^{6} + \frac{1037656}{26778125} a^{5} - \frac{140013}{2434375} a^{4} - \frac{11435643}{26778125} a^{3} + \frac{1743739}{5355625} a^{2} - \frac{55367}{214225} a + \frac{38273}{214225}$, $\frac{1}{607148020010978392035546875} a^{13} - \frac{9338713655443607662}{607148020010978392035546875} a^{12} + \frac{5064559871192271954133}{6391031789589246231953125} a^{11} - \frac{535419617036492325878768}{607148020010978392035546875} a^{10} - \frac{279461845684728702562626}{24285920800439135681421875} a^{9} - \frac{7359265242378279837062122}{607148020010978392035546875} a^{8} + \frac{149733297648277845913187}{24285920800439135681421875} a^{7} - \frac{1220097321537490261994378}{607148020010978392035546875} a^{6} + \frac{9986654191295909666119292}{121429604002195678407109375} a^{5} + \frac{56064048452437753989052848}{607148020010978392035546875} a^{4} + \frac{15605820921303039905725454}{31955158947946231159765625} a^{3} - \frac{29852192379217247374726158}{121429604002195678407109375} a^{2} - \frac{1026062272649164742601789}{4857184160087827136284375} a + \frac{1909553843517629121186524}{4857184160087827136284375}$

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

Class group and class number

$C_{14}$, which has order $14$ (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:  \( 27258655.2736 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_7$ (as 14T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 14
The 5 conjugacy class representatives for $D_{7}$
Character table for $D_{7}$

Intermediate fields

\(\Q(\sqrt{-6191}) \), 7.1.237291625871.1 x7

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 7 sibling: 7.1.237291625871.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
$151$151.2.1.1$x^{2} - 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.1$x^{2} - 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.1$x^{2} - 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.1$x^{2} - 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.1$x^{2} - 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.1$x^{2} - 151$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.1$x^{2} - 151$$2$$1$$1$$C_2$$[\ ]_{2}$