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Properties

Label 14.0.37798701599...9767.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,6263^{7}$
Root discriminant $79.14$
Ramified prime $6263$
Class number $11$ (GRH)
Class group $[11]$ (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![2105941, -3274534, 3480188, -2873985, 1770111, -807128, 262353, -44720, 318, 3672, -819, 135, 18, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 18*x^12 + 135*x^11 - 819*x^10 + 3672*x^9 + 318*x^8 - 44720*x^7 + 262353*x^6 - 807128*x^5 + 1770111*x^4 - 2873985*x^3 + 3480188*x^2 - 3274534*x + 2105941)
 
gp: K = bnfinit(x^14 - 4*x^13 + 18*x^12 + 135*x^11 - 819*x^10 + 3672*x^9 + 318*x^8 - 44720*x^7 + 262353*x^6 - 807128*x^5 + 1770111*x^4 - 2873985*x^3 + 3480188*x^2 - 3274534*x + 2105941, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} + 18 x^{12} + 135 x^{11} - 819 x^{10} + 3672 x^{9} + 318 x^{8} - 44720 x^{7} + 262353 x^{6} - 807128 x^{5} + 1770111 x^{4} - 2873985 x^{3} + 3480188 x^{2} - 3274534 x + 2105941 \)

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:  \(-377987015999056008227829767=-\,6263^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.14$
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:  $6263$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19} a^{10} + \frac{3}{19} a^{9} - \frac{9}{19} a^{8} - \frac{7}{19} a^{7} - \frac{8}{19} a^{6} - \frac{7}{19} a^{4} - \frac{9}{19} a^{3} + \frac{6}{19} a^{2} - \frac{8}{19} a$, $\frac{1}{19} a^{11} + \frac{1}{19} a^{9} + \frac{1}{19} a^{8} - \frac{6}{19} a^{7} + \frac{5}{19} a^{6} - \frac{7}{19} a^{5} - \frac{7}{19} a^{4} - \frac{5}{19} a^{3} - \frac{7}{19} a^{2} + \frac{5}{19} a$, $\frac{1}{139973} a^{12} - \frac{3298}{139973} a^{11} + \frac{3536}{139973} a^{10} + \frac{33338}{139973} a^{9} - \frac{27671}{139973} a^{8} - \frac{49735}{139973} a^{7} + \frac{65632}{139973} a^{6} + \frac{52662}{139973} a^{5} + \frac{46919}{139973} a^{4} - \frac{7010}{139973} a^{3} - \frac{68616}{139973} a^{2} + \frac{7689}{139973} a - \frac{739}{7367}$, $\frac{1}{9508127752457975972743941946705} a^{13} - \frac{5327739611243661740045873}{1901625550491595194548788389341} a^{12} - \frac{73054490652852291646759981717}{9508127752457975972743941946705} a^{11} + \frac{175620300911157858618338048772}{9508127752457975972743941946705} a^{10} - \frac{1052032118547764457090937726801}{9508127752457975972743941946705} a^{9} + \frac{1767736290440420123385018038878}{9508127752457975972743941946705} a^{8} - \frac{897764229259357924918673224492}{1901625550491595194548788389341} a^{7} + \frac{25436449073038419420413035776}{100085555289031326028883599439} a^{6} - \frac{3118433431769294828331469252562}{9508127752457975972743941946705} a^{5} + \frac{2061598084991338702549158429339}{9508127752457975972743941946705} a^{4} - \frac{2521570402409834590250551369298}{9508127752457975972743941946705} a^{3} + \frac{1447789957956196835514777825728}{9508127752457975972743941946705} a^{2} + \frac{267161842398179271197163421014}{1901625550491595194548788389341} a + \frac{145257712965499687824785003974}{500427776445156630144417997195}$

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

Class group and class number

$C_{11}$, which has order $11$ (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:  \( 4447835.41361 \) (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

\(\Q(\sqrt{-6263}) \), 7.1.245667233447.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.245667233447.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
6263Data not computed