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Properties

Label 14.0.15630176392...5591.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 2557^{7}$
Root discriminant $87.58$
Ramified primes $3, 2557$
Class number $12$ (GRH)
Class group $[12]$ (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![1726875, 439425, 1133790, 1106991, -756401, -7434, 224506, -83854, 15553, -3436, 1003, -172, 28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 28*x^12 - 172*x^11 + 1003*x^10 - 3436*x^9 + 15553*x^8 - 83854*x^7 + 224506*x^6 - 7434*x^5 - 756401*x^4 + 1106991*x^3 + 1133790*x^2 + 439425*x + 1726875)
 
gp: K = bnfinit(x^14 + 28*x^12 - 172*x^11 + 1003*x^10 - 3436*x^9 + 15553*x^8 - 83854*x^7 + 224506*x^6 - 7434*x^5 - 756401*x^4 + 1106991*x^3 + 1133790*x^2 + 439425*x + 1726875, 1)
 

Normalized defining polynomial

\( x^{14} + 28 x^{12} - 172 x^{11} + 1003 x^{10} - 3436 x^{9} + 15553 x^{8} - 83854 x^{7} + 224506 x^{6} - 7434 x^{5} - 756401 x^{4} + 1106991 x^{3} + 1133790 x^{2} + 439425 x + 1726875 \)

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:  \(-1563017639236784352773715591=-\,3^{7}\cdot 2557^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.58$
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:  $3, 2557$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{15} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{2}{5} a$, $\frac{1}{15} a^{6} - \frac{1}{3} a^{4} + \frac{4}{15} a^{2}$, $\frac{1}{45} a^{7} - \frac{1}{45} a^{6} - \frac{1}{9} a^{4} - \frac{16}{45} a^{3} + \frac{2}{15} a^{2} + \frac{1}{3} a$, $\frac{1}{225} a^{8} + \frac{1}{225} a^{7} - \frac{2}{225} a^{6} - \frac{1}{45} a^{5} + \frac{19}{225} a^{4} + \frac{19}{225} a^{3} - \frac{7}{25} a^{2} + \frac{2}{15} a$, $\frac{1}{225} a^{9} + \frac{2}{225} a^{7} + \frac{7}{225} a^{6} - \frac{2}{75} a^{5} + \frac{2}{9} a^{4} - \frac{29}{75} a^{3} + \frac{11}{75} a^{2}$, $\frac{1}{675} a^{10} - \frac{1}{675} a^{9} - \frac{1}{675} a^{8} + \frac{2}{675} a^{7} + \frac{8}{675} a^{6} + \frac{11}{675} a^{5} - \frac{194}{675} a^{4} + \frac{71}{225} a^{3} + \frac{47}{225} a^{2} + \frac{2}{5} a + \frac{1}{3}$, $\frac{1}{10125} a^{11} - \frac{1}{3375} a^{10} - \frac{8}{10125} a^{9} + \frac{16}{10125} a^{8} + \frac{58}{10125} a^{7} + \frac{298}{10125} a^{6} - \frac{59}{3375} a^{5} + \frac{304}{10125} a^{4} + \frac{299}{1125} a^{3} - \frac{137}{675} a^{2} - \frac{1}{3} a + \frac{2}{9}$, $\frac{1}{455625} a^{12} - \frac{11}{455625} a^{11} - \frac{224}{455625} a^{10} + \frac{64}{91125} a^{9} + \frac{7}{91125} a^{8} + \frac{344}{455625} a^{7} - \frac{13436}{455625} a^{6} + \frac{2111}{91125} a^{5} + \frac{75754}{455625} a^{4} + \frac{33494}{151875} a^{3} - \frac{11753}{30375} a^{2} + \frac{736}{2025} a - \frac{2}{81}$, $\frac{1}{12194509913872782271875} a^{13} + \frac{1823143144813318}{12194509913872782271875} a^{12} - \frac{566295944086766543}{12194509913872782271875} a^{11} - \frac{5270753317714914826}{12194509913872782271875} a^{10} - \frac{5330747693212503047}{2438901982774556454375} a^{9} - \frac{21236985625784659066}{12194509913872782271875} a^{8} + \frac{265247770184537668}{221718362070414223125} a^{7} + \frac{137915434586252480761}{12194509913872782271875} a^{6} + \frac{400533893286581744149}{12194509913872782271875} a^{5} + \frac{537797319571748769497}{1354945545985864696875} a^{4} + \frac{1398353710265693004161}{4064836637957594090625} a^{3} + \frac{3692636954799830399}{30109901021908104375} a^{2} + \frac{2028066216966475819}{54197821839434587875} a - \frac{353970088067808533}{722637624525794505}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( 846785501.713 \) (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

\(\Q(\sqrt{-7671}) \), 7.1.451394172711.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.451394172711.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/5.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
2557Data not computed