Properties

Label 14.0.579...744.4
Degree $14$
Signature $[0, 7]$
Discriminant $-5.797\times 10^{21}$
Root discriminant $35.85$
Ramified primes $2, 29$
Class number $7$
Class group $[7]$
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 - 2*x^13 + 2*x^12 + 20*x^11 - 83*x^10 + 12*x^9 + 110*x^8 - 176*x^7 + 2447*x^6 - 3952*x^5 - 2038*x^4 - 5862*x^3 + 9737*x^2 + 11592*x + 13033)
 
gp: K = bnfinit(x^14 - 2*x^13 + 2*x^12 + 20*x^11 - 83*x^10 + 12*x^9 + 110*x^8 - 176*x^7 + 2447*x^6 - 3952*x^5 - 2038*x^4 - 5862*x^3 + 9737*x^2 + 11592*x + 13033, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13033, 11592, 9737, -5862, -2038, -3952, 2447, -176, 110, 12, -83, 20, 2, -2, 1]);
 

\(x^{14} - 2 x^{13} + 2 x^{12} + 20 x^{11} - 83 x^{10} + 12 x^{9} + 110 x^{8} - 176 x^{7} + 2447 x^{6} - 3952 x^{5} - 2038 x^{4} - 5862 x^{3} + 9737 x^{2} + 11592 x + 13033\)  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:  $[0, 7]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-5796901408038404767744\)\(\medspace = -\,2^{14}\cdot 29^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $35.85$
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:  $2, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{12789774316960076723418958513} a^{13} + \frac{3444104340696083908158094412}{12789774316960076723418958513} a^{12} - \frac{4739613857555655773131208150}{12789774316960076723418958513} a^{11} - \frac{4291071350582826000407388714}{12789774316960076723418958513} a^{10} - \frac{2562070588677547455380550548}{12789774316960076723418958513} a^{9} - \frac{3402422768623145219542832622}{12789774316960076723418958513} a^{8} + \frac{3026263519484020038696007711}{12789774316960076723418958513} a^{7} + \frac{4118663202952402147778047183}{12789774316960076723418958513} a^{6} + \frac{3864131047626249020164117198}{12789774316960076723418958513} a^{5} - \frac{4174668285972796715749954853}{12789774316960076723418958513} a^{4} + \frac{3601759206183461787477556134}{12789774316960076723418958513} a^{3} - \frac{697608557525749742143957990}{12789774316960076723418958513} a^{2} - \frac{4261980420607665427919548078}{12789774316960076723418958513} a + \frac{5201595678199828828944356224}{12789774316960076723418958513}$  Toggle raw display

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

Class group and class number

$C_{7}$, which has order $7$

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:  \( \frac{12317228248971}{163719440972021573} a^{13} - \frac{26878028333909}{163719440972021573} a^{12} + \frac{92349634788534}{163719440972021573} a^{11} + \frac{204932961825272}{163719440972021573} a^{10} - \frac{945241267907748}{163719440972021573} a^{9} + \frac{1863823155722795}{163719440972021573} a^{8} - \frac{1854211622966132}{163719440972021573} a^{7} - \frac{4444186303339562}{163719440972021573} a^{6} + \frac{33985587918655108}{163719440972021573} a^{5} - \frac{70680160761132613}{163719440972021573} a^{4} + \frac{120436235845816693}{163719440972021573} a^{3} - \frac{134346294211846567}{163719440972021573} a^{2} - \frac{37553476289553289}{163719440972021573} a - \frac{218804396285973327}{163719440972021573} \) (order $4$)  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:  \( 14379.655158793385 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{7}\cdot 14379.655158793385 \cdot 7}{4\sqrt{5796901408038404767744}}\approx 0.127775515402617$

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

\(\Q(\sqrt{-1}) \)

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

Sibling fields

Degree 14 siblings: 14.0.9745585291264.1, 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 R ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$2$2.14.14.38$x^{14} + 4 x^{13} + 3 x^{12} - 2 x^{11} + 2 x^{10} - 2 x^{8} + 4 x^{6} - 2 x^{5} + 4 x^{3} - 2 x^{2} + 2 x + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
$29$29.7.6.5$x^{7} + 58$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.6$x^{7} - 464$$7$$1$$6$$C_7$$[\ ]_{7}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
1.116.14t1.b.c$1$ $ 2^{2} \cdot 29 $ 14.0.5796901408038404767744.1 $C_{14}$ (as 14T1) $0$ $-1$
1.29.7t1.a.e$1$ $ 29 $ 7.7.594823321.1 $C_7$ (as 7T1) $0$ $1$
1.116.14t1.b.d$1$ $ 2^{2} \cdot 29 $ 14.0.5796901408038404767744.1 $C_{14}$ (as 14T1) $0$ $-1$
1.29.7t1.a.f$1$ $ 29 $ 7.7.594823321.1 $C_7$ (as 7T1) $0$ $1$
1.116.14t1.b.e$1$ $ 2^{2} \cdot 29 $ 14.0.5796901408038404767744.1 $C_{14}$ (as 14T1) $0$ $-1$
1.116.14t1.b.b$1$ $ 2^{2} \cdot 29 $ 14.0.5796901408038404767744.1 $C_{14}$ (as 14T1) $0$ $-1$
1.116.14t1.b.a$1$ $ 2^{2} \cdot 29 $ 14.0.5796901408038404767744.1 $C_{14}$ (as 14T1) $0$ $-1$
1.29.7t1.a.b$1$ $ 29 $ 7.7.594823321.1 $C_7$ (as 7T1) $0$ $1$
1.116.14t1.b.f$1$ $ 2^{2} \cdot 29 $ 14.0.5796901408038404767744.1 $C_{14}$ (as 14T1) $0$ $-1$
1.29.7t1.a.c$1$ $ 29 $ 7.7.594823321.1 $C_7$ (as 7T1) $0$ $1$
1.29.7t1.a.d$1$ $ 29 $ 7.7.594823321.1 $C_7$ (as 7T1) $0$ $1$
1.29.7t1.a.a$1$ $ 29 $ 7.7.594823321.1 $C_7$ (as 7T1) $0$ $1$
2.3364.7t2.a.c$2$ $ 2^{2} \cdot 29^{2}$ 7.1.38068692544.1 $D_{7}$ (as 7T2) $1$ $0$
2.3364.14t8.a.c$2$ $ 2^{2} \cdot 29^{2}$ 14.0.5796901408038404767744.4 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.3364.14t8.a.b$2$ $ 2^{2} \cdot 29^{2}$ 14.0.5796901408038404767744.4 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.3364.14t8.a.e$2$ $ 2^{2} \cdot 29^{2}$ 14.0.5796901408038404767744.4 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.116.14t8.a.d$2$ $ 2^{2} \cdot 29 $ 14.0.5796901408038404767744.4 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.116.14t8.a.f$2$ $ 2^{2} \cdot 29 $ 14.0.5796901408038404767744.4 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.3364.14t8.a.a$2$ $ 2^{2} \cdot 29^{2}$ 14.0.5796901408038404767744.4 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.3364.7t2.a.b$2$ $ 2^{2} \cdot 29^{2}$ 7.1.38068692544.1 $D_{7}$ (as 7T2) $1$ $0$
2.3364.14t8.a.f$2$ $ 2^{2} \cdot 29^{2}$ 14.0.5796901408038404767744.4 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.116.14t8.a.e$2$ $ 2^{2} \cdot 29 $ 14.0.5796901408038404767744.4 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.116.14t8.a.a$2$ $ 2^{2} \cdot 29 $ 14.0.5796901408038404767744.4 $C_7 \wr C_2$ (as 14T8) $0$ $0$
* 2.3364.14t8.b.e$2$ $ 2^{2} \cdot 29^{2}$ 14.0.5796901408038404767744.4 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.116.14t8.a.b$2$ $ 2^{2} \cdot 29 $ 14.0.5796901408038404767744.4 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.116.14t8.a.c$2$ $ 2^{2} \cdot 29 $ 14.0.5796901408038404767744.4 $C_7 \wr C_2$ (as 14T8) $0$ $0$
* 2.3364.14t8.b.b$2$ $ 2^{2} \cdot 29^{2}$ 14.0.5796901408038404767744.4 $C_7 \wr C_2$ (as 14T8) $0$ $0$
* 2.3364.14t8.b.c$2$ $ 2^{2} \cdot 29^{2}$ 14.0.5796901408038404767744.4 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.3364.7t2.a.a$2$ $ 2^{2} \cdot 29^{2}$ 7.1.38068692544.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.3364.14t8.b.d$2$ $ 2^{2} \cdot 29^{2}$ 14.0.5796901408038404767744.4 $C_7 \wr C_2$ (as 14T8) $0$ $0$
* 2.3364.14t8.b.f$2$ $ 2^{2} \cdot 29^{2}$ 14.0.5796901408038404767744.4 $C_7 \wr C_2$ (as 14T8) $0$ $0$
* 2.3364.14t8.b.a$2$ $ 2^{2} \cdot 29^{2}$ 14.0.5796901408038404767744.4 $C_7 \wr C_2$ (as 14T8) $0$ $0$
2.3364.14t8.a.d$2$ $ 2^{2} \cdot 29^{2}$ 14.0.5796901408038404767744.4 $C_7 \wr C_2$ (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 *.