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Properties

Label 14.2.22713982309...0000.1
Degree $14$
Signature $[2, 6]$
Discriminant $2^{12}\cdot 3^{12}\cdot 5^{7}\cdot 7^{14}\cdot 11^{12}\cdot 13^{7}$
Root discriminant $2047.22$
Ramified primes $2, 3, 5, 7, 11, 13$
Class number $336$ (GRH)
Class group $[2, 2, 84]$ (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![-264145324, -127228444, 99193108, 39080692, -15055796, -6404692, 1461292, 520087, -91273, -24661, 3731, 637, -91, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 - 91*x^12 + 637*x^11 + 3731*x^10 - 24661*x^9 - 91273*x^8 + 520087*x^7 + 1461292*x^6 - 6404692*x^5 - 15055796*x^4 + 39080692*x^3 + 99193108*x^2 - 127228444*x - 264145324)
 
gp: K = bnfinit(x^14 - 7*x^13 - 91*x^12 + 637*x^11 + 3731*x^10 - 24661*x^9 - 91273*x^8 + 520087*x^7 + 1461292*x^6 - 6404692*x^5 - 15055796*x^4 + 39080692*x^3 + 99193108*x^2 - 127228444*x - 264145324, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} - 91 x^{12} + 637 x^{11} + 3731 x^{10} - 24661 x^{9} - 91273 x^{8} + 520087 x^{7} + 1461292 x^{6} - 6404692 x^{5} - 15055796 x^{4} + 39080692 x^{3} + 99193108 x^{2} - 127228444 x - 264145324 \)

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:  \(22713982309422460169828377403383719236160000000=2^{12}\cdot 3^{12}\cdot 5^{7}\cdot 7^{14}\cdot 11^{12}\cdot 13^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2047.22$
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:  $2, 3, 5, 7, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{235629149615531665918469576109085697314} a^{13} - \frac{48306963243961818910539138279258962081}{235629149615531665918469576109085697314} a^{12} - \frac{2597975276319013231400246549425052371}{235629149615531665918469576109085697314} a^{11} - \frac{12502139507547855362636871035341133491}{117814574807765832959234788054542848657} a^{10} + \frac{40397331860716508471117936710056347561}{235629149615531665918469576109085697314} a^{9} - \frac{3796454030143731379017437325569884597}{117814574807765832959234788054542848657} a^{8} - \frac{93555894154392484413628747396574722705}{235629149615531665918469576109085697314} a^{7} - \frac{28428927954804282034830924661855497652}{117814574807765832959234788054542848657} a^{6} + \frac{14174372628330844386125482769003111795}{117814574807765832959234788054542848657} a^{5} - \frac{38930238391312005653871014277153926397}{235629149615531665918469576109085697314} a^{4} + \frac{17257215328953894553606723167036872909}{117814574807765832959234788054542848657} a^{3} - \frac{3668953663983602438819728108223393327}{117814574807765832959234788054542848657} a^{2} + \frac{56688261510385458947907903942298734044}{117814574807765832959234788054542848657} a - \frac{6498587391031965342965303124675386735}{117814574807765832959234788054542848657}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{84}$, which has order $336$ (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:  $7$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4935923832024880.0 \) (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

\(\Q(\sqrt{65}) \), 7.1.68069081958026688.23

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 R R R R R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
$3$3.14.12.1$x^{14} - 3 x^{7} + 18$$7$$2$$12$$F_7$$[\ ]_{7}^{6}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.7.7.5$x^{7} + 7 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
7.7.7.5$x^{7} + 7 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
$11$11.14.12.1$x^{14} - 11 x^{7} + 847$$7$$2$$12$$(C_7:C_3) \times C_2$$[\ ]_{7}^{6}$
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$