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Properties

Label 14.0.12500336277...1207.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,743^{7}$
Root discriminant $27.26$
Ramified prime $743$
Class number $3$
Class group $[3]$
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![25, 170, 1067, 1127, 3438, 1744, 2679, 608, 519, 108, 80, 8, 15, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 15*x^12 + 8*x^11 + 80*x^10 + 108*x^9 + 519*x^8 + 608*x^7 + 2679*x^6 + 1744*x^5 + 3438*x^4 + 1127*x^3 + 1067*x^2 + 170*x + 25)
 
gp: K = bnfinit(x^14 - 2*x^13 + 15*x^12 + 8*x^11 + 80*x^10 + 108*x^9 + 519*x^8 + 608*x^7 + 2679*x^6 + 1744*x^5 + 3438*x^4 + 1127*x^3 + 1067*x^2 + 170*x + 25, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} + 15 x^{12} + 8 x^{11} + 80 x^{10} + 108 x^{9} + 519 x^{8} + 608 x^{7} + 2679 x^{6} + 1744 x^{5} + 3438 x^{4} + 1127 x^{3} + 1067 x^{2} + 170 x + 25 \)

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:  \(-125003362773881021207=-\,743^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.26$
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:  $743$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{265} a^{11} - \frac{16}{265} a^{10} + \frac{3}{53} a^{9} + \frac{1}{5} a^{8} + \frac{1}{53} a^{7} - \frac{26}{53} a^{6} + \frac{98}{265} a^{4} + \frac{12}{53} a^{3} + \frac{46}{265} a^{2} + \frac{33}{265} a + \frac{26}{53}$, $\frac{1}{8215} a^{12} - \frac{8}{8215} a^{11} - \frac{118}{1643} a^{10} + \frac{130}{1643} a^{9} - \frac{2857}{8215} a^{8} + \frac{3302}{8215} a^{7} - \frac{404}{8215} a^{6} + \frac{2907}{8215} a^{5} - \frac{1276}{8215} a^{4} + \frac{2222}{8215} a^{3} - \frac{553}{8215} a^{2} + \frac{1613}{8215} a + \frac{49}{1643}$, $\frac{1}{1612561722942365} a^{13} - \frac{7282489066}{1612561722942365} a^{12} + \frac{2524670242079}{1612561722942365} a^{11} + \frac{30636199306652}{322512344588473} a^{10} + \frac{144055760152108}{1612561722942365} a^{9} + \frac{420642674394563}{1612561722942365} a^{8} - \frac{112374439091896}{322512344588473} a^{7} - \frac{13407778875856}{1612561722942365} a^{6} + \frac{576605494490838}{1612561722942365} a^{5} + \frac{22080262265105}{322512344588473} a^{4} + \frac{39468648071501}{1612561722942365} a^{3} - \frac{777027961867818}{1612561722942365} a^{2} - \frac{536590891449494}{1612561722942365} a - \frac{27046170444524}{322512344588473}$

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

Class group and class number

$C_{3}$, which has order $3$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6875.76275571 \)
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

\(\Q(\sqrt{-743}) \), 7.1.410172407.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.410172407.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
743Data not computed