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Properties

Label 14.8.77379293087...2667.1
Degree $14$
Signature $[8, 3]$
Discriminant $-\,3^{7}\cdot 29^{12}$
Root discriminant $31.05$
Ramified primes $3, 29$
Class number $1$
Class group Trivial
Galois group $C_2\times F_8$ (as 14T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-17, 61, 158, -175, -343, 104, 365, 119, -289, -104, 114, 28, -18, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^13 - 18*x^12 + 28*x^11 + 114*x^10 - 104*x^9 - 289*x^8 + 119*x^7 + 365*x^6 + 104*x^5 - 343*x^4 - 175*x^3 + 158*x^2 + 61*x - 17)
 
gp: K = bnfinit(x^14 - 3*x^13 - 18*x^12 + 28*x^11 + 114*x^10 - 104*x^9 - 289*x^8 + 119*x^7 + 365*x^6 + 104*x^5 - 343*x^4 - 175*x^3 + 158*x^2 + 61*x - 17, 1)
 

Normalized defining polynomial

\( x^{14} - 3 x^{13} - 18 x^{12} + 28 x^{11} + 114 x^{10} - 104 x^{9} - 289 x^{8} + 119 x^{7} + 365 x^{6} + 104 x^{5} - 343 x^{4} - 175 x^{3} + 158 x^{2} + 61 x - 17 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-773792930870360792667=-\,3^{7}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.05$
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:  $3, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{110048077219} a^{13} - \frac{14097060408}{110048077219} a^{12} - \frac{19874116490}{110048077219} a^{11} + \frac{31581945210}{110048077219} a^{10} + \frac{26681220589}{110048077219} a^{9} + \frac{25309052983}{110048077219} a^{8} + \frac{4465643773}{110048077219} a^{7} + \frac{6540125455}{110048077219} a^{6} + \frac{40487347460}{110048077219} a^{5} - \frac{37406470465}{110048077219} a^{4} - \frac{12050893320}{110048077219} a^{3} + \frac{36403128476}{110048077219} a^{2} - \frac{23110121967}{110048077219} a - \frac{1738875400}{6473416307}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 232028.906801 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_8$ (as 14T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 112
The 16 conjugacy class representatives for $C_2\times F_8$
Character table for $C_2\times F_8$

Intermediate fields

7.7.594823321.1

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

Sibling fields

Degree 16 sibling: data not computed
Degree 28 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.14.0.1}{14} }$ R ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$3$3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$