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Properties

Label 14.0.15127733925...4375.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,5^{7}\cdot 7^{7}\cdot 157^{7}$
Root discriminant $74.13$
Ramified primes $5, 7, 157$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group $D_{7}$ (as 14T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![445655, 1089515, 1667351, 1094380, 562767, 30960, -47811, -35365, -434, 3185, 709, -130, -38, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 38*x^12 - 130*x^11 + 709*x^10 + 3185*x^9 - 434*x^8 - 35365*x^7 - 47811*x^6 + 30960*x^5 + 562767*x^4 + 1094380*x^3 + 1667351*x^2 + 1089515*x + 445655)
 
gp: K = bnfinit(x^14 - 38*x^12 - 130*x^11 + 709*x^10 + 3185*x^9 - 434*x^8 - 35365*x^7 - 47811*x^6 + 30960*x^5 + 562767*x^4 + 1094380*x^3 + 1667351*x^2 + 1089515*x + 445655, 1)
 

Normalized defining polynomial

\( x^{14} - 38 x^{12} - 130 x^{11} + 709 x^{10} + 3185 x^{9} - 434 x^{8} - 35365 x^{7} - 47811 x^{6} + 30960 x^{5} + 562767 x^{4} + 1094380 x^{3} + 1667351 x^{2} + 1089515 x + 445655 \)

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:  \(-151277339258832781070234375=-\,5^{7}\cdot 7^{7}\cdot 157^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.13$
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:  $5, 7, 157$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{17} a^{9} - \frac{5}{17} a^{8} - \frac{5}{17} a^{7} + \frac{4}{17} a^{6} - \frac{6}{17} a^{5} + \frac{6}{17} a^{4} + \frac{4}{17} a^{3} - \frac{5}{17} a^{2} + \frac{6}{17} a$, $\frac{1}{595} a^{10} + \frac{1}{119} a^{9} - \frac{4}{17} a^{8} + \frac{1}{119} a^{7} + \frac{9}{35} a^{6} + \frac{47}{119} a^{5} - \frac{4}{17} a^{4} + \frac{41}{119} a^{3} + \frac{296}{595} a^{2} - \frac{8}{17} a$, $\frac{1}{595} a^{11} + \frac{2}{119} a^{9} - \frac{2}{7} a^{8} - \frac{152}{595} a^{7} + \frac{2}{7} a^{6} + \frac{3}{119} a^{5} + \frac{2}{7} a^{4} - \frac{29}{595} a^{3} - \frac{3}{7} a^{2} + \frac{2}{17} a$, $\frac{1}{3390497425} a^{12} - \frac{9610}{19374271} a^{11} - \frac{214889}{3390497425} a^{10} - \frac{3996557}{678099485} a^{9} + \frac{519108}{3390497425} a^{8} + \frac{23838882}{96871355} a^{7} + \frac{161637419}{484356775} a^{6} + \frac{23502477}{135619897} a^{5} + \frac{754074761}{3390497425} a^{4} + \frac{276154232}{678099485} a^{3} - \frac{1527535994}{3390497425} a^{2} - \frac{24410026}{96871355} a - \frac{514946}{5698315}$, $\frac{1}{13999303844536962721525} a^{13} + \frac{274194244154}{1999900549219566103075} a^{12} - \frac{2588606757710925584}{13999303844536962721525} a^{11} + \frac{5232158646867713753}{13999303844536962721525} a^{10} + \frac{145957621225845394478}{13999303844536962721525} a^{9} - \frac{812717469824230438283}{1999900549219566103075} a^{8} - \frac{839699984307125492781}{1999900549219566103075} a^{7} + \frac{64034309019934840089}{13999303844536962721525} a^{6} - \frac{2105609483123024364389}{13999303844536962721525} a^{5} - \frac{5716461085199665906057}{13999303844536962721525} a^{4} + \frac{410722348808304116691}{13999303844536962721525} a^{3} - \frac{401750937254925348116}{1999900549219566103075} a^{2} + \frac{16800807214672331953}{79996021968782644123} a + \frac{1047973287372293771}{3361177393646329585}$

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:  \( 23628085.1406 \) (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

\(\Q(\sqrt{-5495}) \), 7.1.165921662375.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.165921662375.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ R R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
$157$157.2.1.1$x^{2} - 157$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.1$x^{2} - 157$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.1$x^{2} - 157$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.1$x^{2} - 157$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.1$x^{2} - 157$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.1$x^{2} - 157$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.1$x^{2} - 157$$2$$1$$1$$C_2$$[\ ]_{2}$