Properties

Label 14.2.890987630226941.1
Degree $14$
Signature $[2, 6]$
Discriminant $8.910\times 10^{14}$
Root discriminant $11.69$
Ramified primes $13, 17, 109$
Class number $1$
Class group trivial
Galois group 14T51

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Show commands: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 2*x^12 - 6*x^10 + x^9 + 4*x^8 - 13*x^7 + 4*x^6 + x^5 - 6*x^4 + 2*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^14 - 2*x^13 + 2*x^12 - 6*x^10 + x^9 + 4*x^8 - 13*x^7 + 4*x^6 + x^5 - 6*x^4 + 2*x^2 - 2*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 2, 0, -6, 1, 4, -13, 4, 1, -6, 0, 2, -2, 1]);
 

\(x^{14} - 2 x^{13} + 2 x^{12} - 6 x^{10} + x^{9} + 4 x^{8} - 13 x^{7} + 4 x^{6} + x^{5} - 6 x^{4} + 2 x^{2} - 2 x + 1\)  Toggle raw display

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

Invariants

Degree:  $14$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(890987630226941\)\(\medspace = 13^{5}\cdot 17\cdot 109^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $11.69$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $13, 17, 109$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{181} a^{12} - \frac{69}{181} a^{11} - \frac{82}{181} a^{10} - \frac{48}{181} a^{9} + \frac{34}{181} a^{8} - \frac{57}{181} a^{7} - \frac{12}{181} a^{6} - \frac{57}{181} a^{5} + \frac{34}{181} a^{4} - \frac{48}{181} a^{3} - \frac{82}{181} a^{2} - \frac{69}{181} a + \frac{1}{181}$, $\frac{1}{181} a^{13} + \frac{44}{181} a^{11} + \frac{86}{181} a^{10} - \frac{20}{181} a^{9} - \frac{64}{181} a^{8} + \frac{37}{181} a^{7} + \frac{20}{181} a^{6} + \frac{83}{181} a^{5} - \frac{55}{181} a^{4} + \frac{45}{181} a^{3} + \frac{65}{181} a^{2} - \frac{54}{181} a + \frac{69}{181}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 48.0066041005 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{6}\cdot 48.0066041005 \cdot 1}{2\sqrt{890987630226941}}\approx 0.197913014752$

Galois group

14T51:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 21504
The 48 conjugacy class representatives for [2^7]L(7)=2wrL(7)
Character table for [2^7]L(7)=2wrL(7) is not computed

Intermediate fields

7.3.2007889.2

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

Sibling fields

Degree 14 sibling: data not computed
Degree 28 siblings: data not computed
Degree 42 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 ${\href{/padicField/2.14.0.1}{14} }$ ${\href{/padicField/3.14.0.1}{14} }$ ${\href{/padicField/5.7.0.1}{7} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ R R ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ ${\href{/padicField/23.14.0.1}{14} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ ${\href{/padicField/43.7.0.1}{7} }^{2}$ ${\href{/padicField/47.14.0.1}{14} }$ ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
$109$$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.8.4.1$x^{8} + 712860 x^{4} - 1295029 x^{2} + 127042344900$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$