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Properties

Label 14.0.22439994995...7343.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 29^{13}$
Root discriminant $39.49$
Ramified primes $3, 29$
Class number $48$
Class group $[2, 2, 2, 6]$
Galois group $C_{14}$ (as 14T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1681, -3516, 2377, 665, 1556, -503, 65, -315, 38, -84, 66, -17, 16, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 16*x^12 - 17*x^11 + 66*x^10 - 84*x^9 + 38*x^8 - 315*x^7 + 65*x^6 - 503*x^5 + 1556*x^4 + 665*x^3 + 2377*x^2 - 3516*x + 1681)
 
gp: K = bnfinit(x^14 - x^13 + 16*x^12 - 17*x^11 + 66*x^10 - 84*x^9 + 38*x^8 - 315*x^7 + 65*x^6 - 503*x^5 + 1556*x^4 + 665*x^3 + 2377*x^2 - 3516*x + 1681, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 16 x^{12} - 17 x^{11} + 66 x^{10} - 84 x^{9} + 38 x^{8} - 315 x^{7} + 65 x^{6} - 503 x^{5} + 1556 x^{4} + 665 x^{3} + 2377 x^{2} - 3516 x + 1681 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-22439994995240462987343=-\,3^{7}\cdot 29^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.49$
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 Galois and abelian over $\Q$.
Conductor:  \(87=3\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{87}(1,·)$, $\chi_{87}(35,·)$, $\chi_{87}(5,·)$, $\chi_{87}(38,·)$, $\chi_{87}(71,·)$, $\chi_{87}(7,·)$, $\chi_{87}(80,·)$, $\chi_{87}(49,·)$, $\chi_{87}(82,·)$, $\chi_{87}(52,·)$, $\chi_{87}(86,·)$, $\chi_{87}(25,·)$, $\chi_{87}(62,·)$, $\chi_{87}(16,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{17} a^{9} - \frac{6}{17} a^{8} - \frac{7}{17} a^{7} - \frac{7}{17} a^{6} + \frac{5}{17} a^{5} + \frac{6}{17} a^{4} - \frac{6}{17} a^{3} - \frac{6}{17} a^{2} - \frac{2}{17} a + \frac{4}{17}$, $\frac{1}{17} a^{10} + \frac{8}{17} a^{8} + \frac{2}{17} a^{7} - \frac{3}{17} a^{6} + \frac{2}{17} a^{5} - \frac{4}{17} a^{4} - \frac{8}{17} a^{3} - \frac{4}{17} a^{2} - \frac{8}{17} a + \frac{7}{17}$, $\frac{1}{17} a^{11} - \frac{1}{17} a^{8} + \frac{2}{17} a^{7} + \frac{7}{17} a^{6} + \frac{7}{17} a^{5} - \frac{5}{17} a^{4} - \frac{7}{17} a^{3} + \frac{6}{17} a^{2} + \frac{6}{17} a + \frac{2}{17}$, $\frac{1}{697} a^{12} - \frac{12}{697} a^{11} + \frac{9}{697} a^{10} - \frac{6}{697} a^{9} - \frac{258}{697} a^{8} + \frac{308}{697} a^{7} - \frac{1}{697} a^{6} + \frac{57}{697} a^{5} - \frac{13}{697} a^{4} + \frac{286}{697} a^{3} + \frac{217}{697} a^{2} + \frac{344}{697} a - \frac{7}{17}$, $\frac{1}{31584993917493479} a^{13} + \frac{856902660014}{31584993917493479} a^{12} + \frac{910220185274448}{31584993917493479} a^{11} - \frac{18054269152595}{770365705304719} a^{10} - \frac{681555880715611}{31584993917493479} a^{9} + \frac{989312706677007}{31584993917493479} a^{8} - \frac{5890871095340449}{31584993917493479} a^{7} - \frac{116597290667229}{31584993917493479} a^{6} + \frac{7488257435364579}{31584993917493479} a^{5} - \frac{8711982463399030}{31584993917493479} a^{4} - \frac{10183983355220626}{31584993917493479} a^{3} + \frac{13189889435689658}{31584993917493479} a^{2} - \frac{15403520131820925}{31584993917493479} a + \frac{333489547292032}{770365705304719}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}$, which has order $48$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $6$
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:  \( 6020.98510015 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{14}$ (as 14T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{-87}) \), 7.7.594823321.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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

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$3.14.7.1$x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.13.11$x^{14} + 3712$$14$$1$$13$$C_{14}$$[\ ]_{14}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.3_29.2t1.1c1$1$ $ 3 \cdot 29 $ $x^{2} - x + 22$ $C_2$ (as 2T1) $1$ $-1$
* 1.29.7t1.1c1$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.3_29.14t1.1c1$1$ $ 3 \cdot 29 $ $x^{14} - x^{13} + 16 x^{12} - 17 x^{11} + 66 x^{10} - 84 x^{9} + 38 x^{8} - 315 x^{7} + 65 x^{6} - 503 x^{5} + 1556 x^{4} + 665 x^{3} + 2377 x^{2} - 3516 x + 1681$ $C_{14}$ (as 14T1) $0$ $-1$
* 1.29.7t1.1c2$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.3_29.14t1.1c2$1$ $ 3 \cdot 29 $ $x^{14} - x^{13} + 16 x^{12} - 17 x^{11} + 66 x^{10} - 84 x^{9} + 38 x^{8} - 315 x^{7} + 65 x^{6} - 503 x^{5} + 1556 x^{4} + 665 x^{3} + 2377 x^{2} - 3516 x + 1681$ $C_{14}$ (as 14T1) $0$ $-1$
* 1.29.7t1.1c3$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.3_29.14t1.1c3$1$ $ 3 \cdot 29 $ $x^{14} - x^{13} + 16 x^{12} - 17 x^{11} + 66 x^{10} - 84 x^{9} + 38 x^{8} - 315 x^{7} + 65 x^{6} - 503 x^{5} + 1556 x^{4} + 665 x^{3} + 2377 x^{2} - 3516 x + 1681$ $C_{14}$ (as 14T1) $0$ $-1$
* 1.29.7t1.1c4$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.3_29.14t1.1c4$1$ $ 3 \cdot 29 $ $x^{14} - x^{13} + 16 x^{12} - 17 x^{11} + 66 x^{10} - 84 x^{9} + 38 x^{8} - 315 x^{7} + 65 x^{6} - 503 x^{5} + 1556 x^{4} + 665 x^{3} + 2377 x^{2} - 3516 x + 1681$ $C_{14}$ (as 14T1) $0$ $-1$
* 1.29.7t1.1c5$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.3_29.14t1.1c5$1$ $ 3 \cdot 29 $ $x^{14} - x^{13} + 16 x^{12} - 17 x^{11} + 66 x^{10} - 84 x^{9} + 38 x^{8} - 315 x^{7} + 65 x^{6} - 503 x^{5} + 1556 x^{4} + 665 x^{3} + 2377 x^{2} - 3516 x + 1681$ $C_{14}$ (as 14T1) $0$ $-1$
* 1.29.7t1.1c6$1$ $ 29 $ $x^{7} - x^{6} - 12 x^{5} + 7 x^{4} + 28 x^{3} - 14 x^{2} - 9 x - 1$ $C_7$ (as 7T1) $0$ $1$
* 1.3_29.14t1.1c6$1$ $ 3 \cdot 29 $ $x^{14} - x^{13} + 16 x^{12} - 17 x^{11} + 66 x^{10} - 84 x^{9} + 38 x^{8} - 315 x^{7} + 65 x^{6} - 503 x^{5} + 1556 x^{4} + 665 x^{3} + 2377 x^{2} - 3516 x + 1681$ $C_{14}$ (as 14T1) $0$ $-1$

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 *.