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Properties

Label 14.0.57858073806...0000.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{12}\cdot 5^{12}\cdot 7^{14}\cdot 11^{12}\cdot 43^{7}$
Root discriminant $2579.87$
Ramified primes $2, 5, 7, 11, 43$
Class number $392$ (GRH)
Class group $[2, 14, 14]$ (GRH)
Galois group $F_7 \times C_2$ (as 14T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![30267061, -26274787, 1782088, 1151073, 6321161, -2259796, 796257, -209793, 73157, -14266, 3731, -497, 98, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 98*x^12 - 497*x^11 + 3731*x^10 - 14266*x^9 + 73157*x^8 - 209793*x^7 + 796257*x^6 - 2259796*x^5 + 6321161*x^4 + 1151073*x^3 + 1782088*x^2 - 26274787*x + 30267061)
 
gp: K = bnfinit(x^14 - 7*x^13 + 98*x^12 - 497*x^11 + 3731*x^10 - 14266*x^9 + 73157*x^8 - 209793*x^7 + 796257*x^6 - 2259796*x^5 + 6321161*x^4 + 1151073*x^3 + 1782088*x^2 - 26274787*x + 30267061, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 98 x^{12} - 497 x^{11} + 3731 x^{10} - 14266 x^{9} + 73157 x^{8} - 209793 x^{7} + 796257 x^{6} - 2259796 x^{5} + 6321161 x^{4} + 1151073 x^{3} + 1782088 x^{2} - 26274787 x + 30267061 \)

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:  \(-578580738069469566208904832694868803000000000000=-\,2^{12}\cdot 5^{12}\cdot 7^{14}\cdot 11^{12}\cdot 43^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2579.87$
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:  $2, 5, 7, 11, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{11} a^{8} - \frac{4}{11} a^{7} - \frac{5}{11} a^{6} - \frac{4}{11} a^{5} + \frac{1}{11} a^{4}$, $\frac{1}{11} a^{9} + \frac{1}{11} a^{7} - \frac{2}{11} a^{6} - \frac{4}{11} a^{5} + \frac{4}{11} a^{4}$, $\frac{1}{11} a^{10} + \frac{2}{11} a^{7} + \frac{1}{11} a^{6} - \frac{3}{11} a^{5} - \frac{1}{11} a^{4}$, $\frac{1}{11} a^{11} - \frac{2}{11} a^{7} - \frac{4}{11} a^{6} - \frac{4}{11} a^{5} - \frac{2}{11} a^{4}$, $\frac{1}{11} a^{12} - \frac{1}{11} a^{7} - \frac{3}{11} a^{6} + \frac{1}{11} a^{5} + \frac{2}{11} a^{4}$, $\frac{1}{206670901530689000396978329039042973} a^{13} + \frac{4302463571193415346896020437927661}{206670901530689000396978329039042973} a^{12} - \frac{5634215500145249577394495395587689}{206670901530689000396978329039042973} a^{11} + \frac{2905862873065135093628563466082525}{206670901530689000396978329039042973} a^{10} - \frac{5231997757947289318151867711176931}{206670901530689000396978329039042973} a^{9} + \frac{4318852876663824441799441031097707}{206670901530689000396978329039042973} a^{8} + \frac{86450459668249219091296104020476970}{206670901530689000396978329039042973} a^{7} - \frac{9163737554544729146028523095411204}{18788263775517181854270757185367543} a^{6} + \frac{2368603815098782555240115835954934}{206670901530689000396978329039042973} a^{5} + \frac{15003527496807250978040768451253664}{206670901530689000396978329039042973} a^{4} + \frac{8630674095474825241930607391724504}{18788263775517181854270757185367543} a^{3} + \frac{8228561121864609163644146257143274}{18788263775517181854270757185367543} a^{2} - \frac{4055801571758165106144268310836803}{18788263775517181854270757185367543} a - \frac{2965516408147909607134670226213571}{18788263775517181854270757185367543}$

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

Class group and class number

$C_{2}\times C_{14}\times C_{14}$, which has order $392$ (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:  \( 6148101058884126.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_7$ (as 14T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 84
The 14 conjugacy class representatives for $F_7 \times C_2$
Character table for $F_7 \times C_2$

Intermediate fields

\(\Q(\sqrt{-43}) \), 7.1.1458956660623000000.11

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

Sibling fields

Degree 14 sibling: data not computed
Degree 28 sibling: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R R R ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.14.12.1$x^{14} - 2 x^{7} + 4$$7$$2$$12$$(C_7:C_3) \times C_2$$[\ ]_{7}^{6}$
$5$5.14.12.1$x^{14} - 5 x^{7} + 50$$7$$2$$12$$F_7$$[\ ]_{7}^{6}$
$7$7.14.14.21$x^{14} + 28 x^{12} + 42 x^{11} + 42 x^{9} + 21 x^{8} + 29 x^{7} + 21 x^{6} + 35 x^{5} + 7 x^{4} + 14 x^{3} + 28 x^{2} + 42 x + 45$$7$$2$$14$$F_7 \times C_2$$[7/6]_{6}^{2}$
$11$11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
$43$43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$