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Properties

Label 14.0.33733763135...0704.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{6}\cdot 7^{24}\cdot 31^{7}$
Root discriminant $210.59$
Ramified primes $2, 7, 31$
Class number $21$ (GRH)
Class group $[21]$ (GRH)
Galois group $C_7^2:S_3$ (as 14T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![336238208, 348280352, 197804880, 95169270, 42847770, 12260920, 570164, -607634, -94150, 19838, 5271, -147, -105, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 105*x^12 - 147*x^11 + 5271*x^10 + 19838*x^9 - 94150*x^8 - 607634*x^7 + 570164*x^6 + 12260920*x^5 + 42847770*x^4 + 95169270*x^3 + 197804880*x^2 + 348280352*x + 336238208)
 
gp: K = bnfinit(x^14 - 105*x^12 - 147*x^11 + 5271*x^10 + 19838*x^9 - 94150*x^8 - 607634*x^7 + 570164*x^6 + 12260920*x^5 + 42847770*x^4 + 95169270*x^3 + 197804880*x^2 + 348280352*x + 336238208, 1)
 

Normalized defining polynomial

\( x^{14} - 105 x^{12} - 147 x^{11} + 5271 x^{10} + 19838 x^{9} - 94150 x^{8} - 607634 x^{7} + 570164 x^{6} + 12260920 x^{5} + 42847770 x^{4} + 95169270 x^{3} + 197804880 x^{2} + 348280352 x + 336238208 \)

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:  \(-337337631352558562817384077600704=-\,2^{6}\cdot 7^{24}\cdot 31^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $210.59$
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:  $2, 7, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{3}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{33943407191346728233461186063293185464615842144} a^{13} + \frac{418516896026689380110060920014211356966879711}{8485851797836682058365296515823296366153960536} a^{12} + \frac{2656200787485609157505739931623691530635460567}{33943407191346728233461186063293185464615842144} a^{11} - \frac{1464119244341282853836090033664022253349808207}{33943407191346728233461186063293185464615842144} a^{10} - \frac{6409657171168562365089327913896554304539221693}{33943407191346728233461186063293185464615842144} a^{9} + \frac{1997762671017796336023611292128022692054870001}{16971703595673364116730593031646592732307921072} a^{8} + \frac{7170768629378271007713670479570270848445891889}{16971703595673364116730593031646592732307921072} a^{7} - \frac{3116102953294730467165865594780414137790941277}{16971703595673364116730593031646592732307921072} a^{6} - \frac{798817690008303404402781009491928548098647273}{8485851797836682058365296515823296366153960536} a^{5} + \frac{764875409373275694671777576076125712959914401}{4242925898918341029182648257911648183076980268} a^{4} - \frac{3326929353782267454856771446113767029925169571}{16971703595673364116730593031646592732307921072} a^{3} + \frac{6809952479109608182236165420321440235138282199}{16971703595673364116730593031646592732307921072} a^{2} + \frac{1823720319701381772860755918221875519470619279}{4242925898918341029182648257911648183076980268} a - \frac{309594565797954548820025255002711920477350052}{1060731474729585257295662064477912045769245067}$

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

Class group and class number

$C_{21}$, which has order $21$ (assuming GRH)

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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 8415941567.41 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7^2:S_3$ (as 14T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 294
The 20 conjugacy class representatives for $C_7^2:S_3$
Character table for $C_7^2:S_3$

Intermediate fields

\(\Q(\sqrt{-31}) \)

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

Sibling fields

Degree 21 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 R ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ R ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
$7$7.7.12.8$x^{7} + 21 x^{6} + 7$$7$$1$$12$$C_7:C_3$$[2]^{3}$
7.7.12.9$x^{7} + 35 x^{6} + 7$$7$$1$$12$$C_7:C_3$$[2]^{3}$
$31$31.14.7.2$x^{14} - 887503681 x^{2} + 495227053998$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$