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Properties

Label 14.0.21472479964...8848.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{20}\cdot 3^{12}\cdot 7^{11}\cdot 11^{7}$
Root discriminant $105.61$
Ramified primes $2, 3, 7, 11$
Class number $112$ (GRH)
Class group $[2, 2, 28]$ (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![12774708, 28047240, 12778236, -8150904, -3601860, -26052, 1093128, -117188, -55280, 948, 2966, -64, -60, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 60*x^12 - 64*x^11 + 2966*x^10 + 948*x^9 - 55280*x^8 - 117188*x^7 + 1093128*x^6 - 26052*x^5 - 3601860*x^4 - 8150904*x^3 + 12778236*x^2 + 28047240*x + 12774708)
 
gp: K = bnfinit(x^14 - 2*x^13 - 60*x^12 - 64*x^11 + 2966*x^10 + 948*x^9 - 55280*x^8 - 117188*x^7 + 1093128*x^6 - 26052*x^5 - 3601860*x^4 - 8150904*x^3 + 12778236*x^2 + 28047240*x + 12774708, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} - 60 x^{12} - 64 x^{11} + 2966 x^{10} + 948 x^{9} - 55280 x^{8} - 117188 x^{7} + 1093128 x^{6} - 26052 x^{5} - 3601860 x^{4} - 8150904 x^{3} + 12778236 x^{2} + 28047240 x + 12774708 \)

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:  \(-21472479964358571500894158848=-\,2^{20}\cdot 3^{12}\cdot 7^{11}\cdot 11^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.61$
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:  $2, 3, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{14} a^{9} - \frac{1}{7} a^{8} - \frac{3}{14} a^{7} - \frac{3}{7} a^{6} + \frac{2}{7} a^{5} + \frac{1}{7} a^{4} - \frac{3}{7} a^{3} - \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{14} a^{10} + \frac{1}{7} a^{7} + \frac{3}{7} a^{6} - \frac{2}{7} a^{5} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3} - \frac{2}{7} a^{2} - \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{42} a^{11} + \frac{1}{42} a^{10} + \frac{1}{21} a^{8} + \frac{4}{21} a^{7} - \frac{2}{7} a^{6} - \frac{10}{21} a^{5} - \frac{1}{3} a^{4} + \frac{2}{7} a^{3} - \frac{1}{7} a^{2} - \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{1034922} a^{12} + \frac{9113}{1034922} a^{11} + \frac{9481}{1034922} a^{10} - \frac{4448}{517461} a^{9} + \frac{7787}{517461} a^{8} + \frac{110176}{517461} a^{7} + \frac{56090}{517461} a^{6} - \frac{136961}{517461} a^{5} + \frac{31310}{73923} a^{4} + \frac{9931}{24641} a^{3} - \frac{78101}{172487} a^{2} + \frac{11523}{24641} a + \frac{83081}{172487}$, $\frac{1}{1566268024540710311838584457652465788} a^{13} - \frac{24663200530393417850614437685}{783134012270355155919292228826232894} a^{12} + \frac{48734224783416913405159932760898}{9550414783784818974625514985685767} a^{11} + \frac{3407841953429952207318587451593647}{261044670756785051973097409608744298} a^{10} - \frac{26323738672109223221202474061985399}{783134012270355155919292228826232894} a^{9} + \frac{2556932842635421507563063027034238}{55938143733596796851378016344730921} a^{8} - \frac{3572376150029134081039751855591244}{18646047911198932283792672114910307} a^{7} + \frac{290948244402539009232461360604594449}{783134012270355155919292228826232894} a^{6} + \frac{30746873953864092325031308770646861}{391567006135177577959646114413116447} a^{5} + \frac{82545891566518324715741597461353670}{391567006135177577959646114413116447} a^{4} - \frac{17018818475084930111545595419942377}{130522335378392525986548704804372149} a^{3} - \frac{20687525090181684544437630982436635}{130522335378392525986548704804372149} a^{2} + \frac{4706344710900875681149847834417139}{18646047911198932283792672114910307} a - \frac{3130432320541884134091310596719287}{18646047911198932283792672114910307}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{28}$, which has order $112$ (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:  \( 13603403.850517571 \) (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

\(\Q(\sqrt{-77}) \), 7.1.784147392.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R R ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.14.20.10$x^{14} + 2 x^{12} + 2 x^{11} + 4 x^{10} + 2 x^{8} - 2 x^{7} - 2 x^{6} + 4 x^{4} + 4 x^{3} - 2$$14$$1$$20$$(C_7:C_3) \times C_2$$[2]_{7}^{3}$
$3$3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$