↑

Properties

Label 14.0.47924308991...6959.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,11^{7}\cdot 19^{7}\cdot 31^{7}$
Root discriminant $80.49$
Ramified primes $11, 19, 31$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group $D_{7}$ (as 14T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12295717, -22475936, 24569154, -12829434, 3860271, -371351, -67254, 11487, 10157, -3213, 214, 104, -6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 - 6*x^12 + 104*x^11 + 214*x^10 - 3213*x^9 + 10157*x^8 + 11487*x^7 - 67254*x^6 - 371351*x^5 + 3860271*x^4 - 12829434*x^3 + 24569154*x^2 - 22475936*x + 12295717)
 
gp: K = bnfinit(x^14 - 4*x^13 - 6*x^12 + 104*x^11 + 214*x^10 - 3213*x^9 + 10157*x^8 + 11487*x^7 - 67254*x^6 - 371351*x^5 + 3860271*x^4 - 12829434*x^3 + 24569154*x^2 - 22475936*x + 12295717, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} - 6 x^{12} + 104 x^{11} + 214 x^{10} - 3213 x^{9} + 10157 x^{8} + 11487 x^{7} - 67254 x^{6} - 371351 x^{5} + 3860271 x^{4} - 12829434 x^{3} + 24569154 x^{2} - 22475936 x + 12295717 \)

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:  \(-479243089919880156320166959=-\,11^{7}\cdot 19^{7}\cdot 31^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.49$
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:  $11, 19, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10} - \frac{2}{11} a^{9} + \frac{4}{11} a^{8} + \frac{3}{11} a^{7} + \frac{5}{11} a^{6} - \frac{1}{11} a^{5} + \frac{1}{11} a^{3} - \frac{3}{11} a^{2} - \frac{1}{11} a + \frac{4}{11}$, $\frac{1}{77} a^{11} - \frac{2}{77} a^{10} + \frac{37}{77} a^{9} + \frac{2}{11} a^{8} - \frac{4}{11} a^{7} + \frac{3}{11} a^{6} - \frac{3}{11} a^{4} + \frac{30}{77} a^{3} + \frac{10}{77} a^{2} + \frac{4}{77} a$, $\frac{1}{1463} a^{12} - \frac{3}{1463} a^{11} - \frac{52}{1463} a^{10} + \frac{390}{1463} a^{9} - \frac{91}{209} a^{8} + \frac{78}{209} a^{7} - \frac{68}{209} a^{6} + \frac{43}{209} a^{5} - \frac{334}{1463} a^{4} + \frac{274}{1463} a^{3} - \frac{580}{1463} a^{2} + \frac{37}{77} a - \frac{1}{11}$, $\frac{1}{205164307884785576943541833495520324007} a^{13} - \frac{68674791279401454876433791695222170}{205164307884785576943541833495520324007} a^{12} - \frac{16747116688929571459186422366951957}{205164307884785576943541833495520324007} a^{11} + \frac{130488219362216617333805719837992932}{10798121467620293523344307026080017053} a^{10} + \frac{98491660763636996074398380295894086046}{205164307884785576943541833495520324007} a^{9} + \frac{10037832627228015263535422985200112500}{29309186840683653849077404785074332001} a^{8} - \frac{8764125078399579239638800651491797470}{29309186840683653849077404785074332001} a^{7} + \frac{6353231121799015029224778497793207352}{29309186840683653849077404785074332001} a^{6} - \frac{50468686260832429226040692020983286048}{205164307884785576943541833495520324007} a^{5} + \frac{21948672839419830918608586502425871}{141395112256916317673012979666106357} a^{4} - \frac{91033176425167305145811048184146833535}{205164307884785576943541833495520324007} a^{3} + \frac{97548279286974227894994986034898992470}{205164307884785576943541833495520324007} a^{2} - \frac{505457715081370594666901348250536686}{10798121467620293523344307026080017053} a - \frac{421395165907717152282906122493205066}{1542588781088613360477758146582859579}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( 6745794.57801 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_7$ (as 14T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 14
The 5 conjugacy class representatives for $D_{7}$
Character table for $D_{7}$

Intermediate fields

\(\Q(\sqrt{-6479}) \), 7.1.271971840239.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.271971840239.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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

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
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
$31$31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$