Properties

Label 14.4.1237706798550547.1
Degree $14$
Signature $[4, 5]$
Discriminant $-1.238\times 10^{15}$
Root discriminant $11.97$
Ramified primes $13, 109, 307$
Class number $1$
Class group trivial
Galois group 14T51

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 3*x^13 + 3*x^12 + 4*x^11 - 14*x^10 + 16*x^9 - 5*x^8 - 6*x^7 + 12*x^6 - 16*x^5 + 17*x^4 - 11*x^3 + 2*x^2 + 2*x - 1)
 
gp: K = bnfinit(x^14 - 3*x^13 + 3*x^12 + 4*x^11 - 14*x^10 + 16*x^9 - 5*x^8 - 6*x^7 + 12*x^6 - 16*x^5 + 17*x^4 - 11*x^3 + 2*x^2 + 2*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 2, 2, -11, 17, -16, 12, -6, -5, 16, -14, 4, 3, -3, 1]);
 

\(x^{14} - 3 x^{13} + 3 x^{12} + 4 x^{11} - 14 x^{10} + 16 x^{9} - 5 x^{8} - 6 x^{7} + 12 x^{6} - 16 x^{5} + 17 x^{4} - 11 x^{3} + 2 x^{2} + 2 x - 1\)  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:  $[4, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-1237706798550547\)\(\medspace = -\,13^{4}\cdot 109^{4}\cdot 307\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $11.97$
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:  $13, 109, 307$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} + \frac{5}{11} a^{11} + \frac{5}{11} a^{10} - \frac{3}{11} a^{9} + \frac{3}{11} a^{8} + \frac{2}{11} a^{6} - \frac{1}{11} a^{5} + \frac{5}{11} a^{4} - \frac{4}{11} a^{3} + \frac{4}{11} a^{2} - \frac{3}{11} a + \frac{2}{11}$, $\frac{1}{121} a^{13} + \frac{1}{121} a^{12} + \frac{7}{121} a^{11} + \frac{32}{121} a^{10} - \frac{7}{121} a^{9} - \frac{12}{121} a^{8} - \frac{53}{121} a^{7} + \frac{24}{121} a^{6} - \frac{13}{121} a^{5} + \frac{53}{121} a^{4} - \frac{13}{121} a^{3} + \frac{58}{121} a^{2} - \frac{8}{121} a - \frac{30}{121}$  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:  $8$
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:  \( 83.4257736319 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{4}\cdot(2\pi)^{5}\cdot 83.4257736319 \cdot 1}{2\sqrt{1237706798550547}}\approx 0.185772182459$

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 ${\href{/padicField/2.14.0.1}{14} }$ ${\href{/padicField/3.14.0.1}{14} }$ ${\href{/padicField/5.14.0.1}{14} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ R ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{3}$ ${\href{/padicField/23.14.0.1}{14} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ ${\href{/padicField/43.14.0.1}{14} }$ ${\href{/padicField/47.14.0.1}{14} }$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$109$$\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}$
$307$Data not computed