Properties

Label 14.0.30270895108587.1
Degree $14$
Signature $[0, 7]$
Discriminant $-3.027\times 10^{13}$
Root discriminant $9.18$
Ramified primes $3, 7$
Class number $1$
Class group trivial
Galois group $C_7 \wr C_2$ (as 14T8)

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Normalized defining polynomial

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

\( x^{14} - 4 x^{13} + 6 x^{12} - 3 x^{11} - 6 x^{10} + 14 x^{9} - 12 x^{8} + 2 x^{7} + 10 x^{6} - 8 x^{5} - 3 x^{4} + x^{3} + 7 x^{2} - 5 x + 1 \)

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

Invariants

Degree:  $14$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 7]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-30270895108587\)\(\medspace = -\,3^{7}\cdot 7^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $9.18$
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:  $3, 7$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $7$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{7289} a^{13} + \frac{681}{7289} a^{12} - \frac{5}{7289} a^{11} - \frac{3428}{7289} a^{10} - \frac{1128}{7289} a^{9} - \frac{32}{7289} a^{8} - \frac{65}{7289} a^{7} - \frac{789}{7289} a^{6} - \frac{1069}{7289} a^{5} - \frac{3373}{7289} a^{4} + \frac{105}{7289} a^{3} - \frac{964}{7289} a^{2} + \frac{2966}{7289} a - \frac{1926}{7289}$

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:  $6$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{315}{197} a^{13} - \frac{1200}{197} a^{12} + \frac{1577}{197} a^{11} - \frac{457}{197} a^{10} - \frac{2099}{197} a^{9} + \frac{3907}{197} a^{8} - \frac{2548}{197} a^{7} - \frac{315}{197} a^{6} + \frac{3287}{197} a^{5} - \frac{1650}{197} a^{4} - \frac{1794}{197} a^{3} - \frac{83}{197} a^{2} + \frac{2283}{197} a - \frac{718}{197} \) (order $6$)
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:  \( 16.520791850817314 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{7}\cdot 16.520791850817314 \cdot 1}{6\sqrt{30270895108587}}\approx 0.193475534880756$

Galois group

$C_7\times D_7$ (as 14T8):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 98
The 35 conjugacy class representatives for $C_7 \wr C_2$
Character table for $C_7 \wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \)

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

Sibling fields

Degree 14 siblings: Deg 14, Deg 14

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 ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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

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
$3$3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$7$7.7.12.2$x^{7} - 7 x^{6} + 56$$7$$1$$12$$C_7$$[2]$
7.7.0.1$x^{7} - x + 2$$1$$7$$0$$C_7$$[\ ]^{7}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
1.147.14t1.a.d$1$ $ 3 \cdot 7^{2}$ 14.0.418988153029298748294987.1 $C_{14}$ (as 14T1) $0$ $-1$
1.147.14t1.a.b$1$ $ 3 \cdot 7^{2}$ 14.0.418988153029298748294987.1 $C_{14}$ (as 14T1) $0$ $-1$
1.147.14t1.a.f$1$ $ 3 \cdot 7^{2}$ 14.0.418988153029298748294987.1 $C_{14}$ (as 14T1) $0$ $-1$
1.147.14t1.a.e$1$ $ 3 \cdot 7^{2}$ 14.0.418988153029298748294987.1 $C_{14}$ (as 14T1) $0$ $-1$
1.147.14t1.a.c$1$ $ 3 \cdot 7^{2}$ 14.0.418988153029298748294987.1 $C_{14}$ (as 14T1) $0$ $-1$
1.49.7t1.a.e$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
1.49.7t1.a.c$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
1.49.7t1.a.b$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
1.49.7t1.a.a$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
1.49.7t1.a.d$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
1.147.14t1.a.a$1$ $ 3 \cdot 7^{2}$ 14.0.418988153029298748294987.1 $C_{14}$ (as 14T1) $0$ $-1$
1.49.7t1.a.f$1$ $ 7^{2}$ 7.7.13841287201.1 $C_7$ (as 7T1) $0$ $1$
* 2.147.14t8.a.b$2$ $ 3 \cdot 7^{2}$ 14.0.30270895108587.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.7203.7t2.a.a$2$ $ 3 \cdot 7^{4}$ 7.1.373714754427.1 $D_{7}$ (as 7T2) $1$ $0$
2.7203.14t8.a.a$2$ $ 3 \cdot 7^{4}$ 14.0.30270895108587.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.7203.14t8.b.c$2$ $ 3 \cdot 7^{4}$ 14.0.30270895108587.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.7203.14t8.a.b$2$ $ 3 \cdot 7^{4}$ 14.0.30270895108587.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
* 2.147.14t8.a.e$2$ $ 3 \cdot 7^{2}$ 14.0.30270895108587.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.7203.14t8.b.b$2$ $ 3 \cdot 7^{4}$ 14.0.30270895108587.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.7203.14t8.a.d$2$ $ 3 \cdot 7^{4}$ 14.0.30270895108587.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.7203.7t2.a.c$2$ $ 3 \cdot 7^{4}$ 7.1.373714754427.1 $D_{7}$ (as 7T2) $1$ $0$
2.7203.14t8.b.f$2$ $ 3 \cdot 7^{4}$ 14.0.30270895108587.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
* 2.147.14t8.a.d$2$ $ 3 \cdot 7^{2}$ 14.0.30270895108587.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
* 2.147.14t8.a.a$2$ $ 3 \cdot 7^{2}$ 14.0.30270895108587.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.7203.14t8.a.e$2$ $ 3 \cdot 7^{4}$ 14.0.30270895108587.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.7203.14t8.b.e$2$ $ 3 \cdot 7^{4}$ 14.0.30270895108587.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.7203.14t8.a.c$2$ $ 3 \cdot 7^{4}$ 14.0.30270895108587.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
* 2.147.14t8.a.c$2$ $ 3 \cdot 7^{2}$ 14.0.30270895108587.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
* 2.147.14t8.a.f$2$ $ 3 \cdot 7^{2}$ 14.0.30270895108587.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.7203.7t2.a.b$2$ $ 3 \cdot 7^{4}$ 7.1.373714754427.1 $D_{7}$ (as 7T2) $1$ $0$
2.7203.14t8.b.d$2$ $ 3 \cdot 7^{4}$ 14.0.30270895108587.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.7203.14t8.a.f$2$ $ 3 \cdot 7^{4}$ 14.0.30270895108587.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.7203.14t8.b.a$2$ $ 3 \cdot 7^{4}$ 14.0.30270895108587.1 $C_7 \wr C_2$ (as 14T8) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.