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Properties

Label 14.2.17382920298...8433.1
Degree $14$
Signature $[2, 6]$
Discriminant $17^{3}\cdot 29^{12}$
Root discriminant $32.90$
Ramified primes $17, 29$
Class number $4$
Class group $[2, 2]$
Galois group $C_2 \wr C_7$ (as 14T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-463, 1212, -784, 298, 1594, -1554, 1413, -460, 46, 172, -94, 26, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 6*x^12 + 26*x^11 - 94*x^10 + 172*x^9 + 46*x^8 - 460*x^7 + 1413*x^6 - 1554*x^5 + 1594*x^4 + 298*x^3 - 784*x^2 + 1212*x - 463)
 
gp: K = bnfinit(x^14 - 4*x^13 + 6*x^12 + 26*x^11 - 94*x^10 + 172*x^9 + 46*x^8 - 460*x^7 + 1413*x^6 - 1554*x^5 + 1594*x^4 + 298*x^3 - 784*x^2 + 1212*x - 463, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} + 6 x^{12} + 26 x^{11} - 94 x^{10} + 172 x^{9} + 46 x^{8} - 460 x^{7} + 1413 x^{6} - 1554 x^{5} + 1594 x^{4} + 298 x^{3} - 784 x^{2} + 1212 x - 463 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1738292029888469398433=17^{3}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.90$
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:  $17, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{118} a^{12} + \frac{2}{59} a^{11} + \frac{2}{59} a^{10} - \frac{19}{118} a^{9} - \frac{14}{59} a^{8} + \frac{2}{59} a^{7} - \frac{16}{59} a^{6} - \frac{13}{59} a^{5} + \frac{51}{118} a^{4} - \frac{13}{59} a^{3} - \frac{53}{118} a^{2} - \frac{9}{118} a + \frac{1}{59}$, $\frac{1}{8985244272733714} a^{13} + \frac{12602664268617}{8985244272733714} a^{12} + \frac{332048197343865}{4492622136366857} a^{11} - \frac{221071110083243}{4492622136366857} a^{10} + \frac{74580874131095}{4492622136366857} a^{9} + \frac{1434668092745233}{8985244272733714} a^{8} - \frac{441211675177906}{4492622136366857} a^{7} - \frac{3242262072576227}{8985244272733714} a^{6} - \frac{492127772189650}{4492622136366857} a^{5} + \frac{1166848550099240}{4492622136366857} a^{4} - \frac{633026139225619}{4492622136366857} a^{3} - \frac{836104789486069}{8985244272733714} a^{2} - \frac{75618398429477}{4492622136366857} a + \frac{1718297675973758}{4492622136366857}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
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:  \( 21387.7113573 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2 \wr C_7$ (as 14T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 896
The 32 conjugacy class representatives for $C_2 \wr C_7$
Character table for $C_2 \wr C_7$ is not computed

Intermediate fields

7.7.594823321.1

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

Sibling fields

Degree 14 siblings: data not computed
Degree 28 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{10}$

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
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
$29$29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$