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Properties

Label 14.0.31247549857...9375.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,5^{7}\cdot 23^{7}\cdot 53^{7}$
Root discriminant $78.07$
Ramified primes $5, 23, 53$
Class number $12$ (GRH)
Class group $[2, 6]$ (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![4158745, 4992495, 1085079, -562305, 662185, 355203, -57037, 867, 11853, -1757, -449, 161, -12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 - 12*x^12 + 161*x^11 - 449*x^10 - 1757*x^9 + 11853*x^8 + 867*x^7 - 57037*x^6 + 355203*x^5 + 662185*x^4 - 562305*x^3 + 1085079*x^2 + 4992495*x + 4158745)
 
gp: K = bnfinit(x^14 - 4*x^13 - 12*x^12 + 161*x^11 - 449*x^10 - 1757*x^9 + 11853*x^8 + 867*x^7 - 57037*x^6 + 355203*x^5 + 662185*x^4 - 562305*x^3 + 1085079*x^2 + 4992495*x + 4158745, 1)
 

Normalized defining polynomial

\( x^{14} - 4 x^{13} - 12 x^{12} + 161 x^{11} - 449 x^{10} - 1757 x^{9} + 11853 x^{8} + 867 x^{7} - 57037 x^{6} + 355203 x^{5} + 662185 x^{4} - 562305 x^{3} + 1085079 x^{2} + 4992495 x + 4158745 \)

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:  \(-312475498577452580635859375=-\,5^{7}\cdot 23^{7}\cdot 53^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.07$
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:  $5, 23, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{725} a^{11} + \frac{27}{725} a^{10} + \frac{18}{725} a^{9} + \frac{13}{725} a^{8} + \frac{189}{725} a^{7} + \frac{62}{145} a^{6} + \frac{194}{725} a^{5} + \frac{253}{725} a^{4} + \frac{293}{725} a^{3} + \frac{267}{725} a^{2} + \frac{8}{29} a - \frac{2}{5}$, $\frac{1}{83375} a^{12} - \frac{1}{16675} a^{11} + \frac{3069}{83375} a^{10} - \frac{5638}{83375} a^{9} - \frac{54}{3625} a^{8} + \frac{9922}{83375} a^{7} + \frac{14489}{83375} a^{6} - \frac{99}{3335} a^{5} + \frac{474}{3625} a^{4} + \frac{21051}{83375} a^{3} + \frac{40811}{83375} a^{2} + \frac{49}{725} a + \frac{3}{25}$, $\frac{1}{1190144010047082711312964740875} a^{13} - \frac{265951299060938072472094}{47605760401883308452518589635} a^{12} + \frac{310157504978921592473420894}{1190144010047082711312964740875} a^{11} - \frac{91390996229901832024807721418}{1190144010047082711312964740875} a^{10} - \frac{39916336847767415883940921657}{1190144010047082711312964740875} a^{9} + \frac{547766191956580609977142126}{8687182555088194973087333875} a^{8} - \frac{161150701266195078513051143976}{1190144010047082711312964740875} a^{7} + \frac{26255435989418066024244494614}{238028802009416542262592948175} a^{6} - \frac{184007589388023460039182721173}{1190144010047082711312964740875} a^{5} + \frac{8859189102951967303894706961}{1190144010047082711312964740875} a^{4} + \frac{95342192737917962355835191966}{1190144010047082711312964740875} a^{3} + \frac{32741555601462918037836295148}{238028802009416542262592948175} a^{2} + \frac{2734506621691526481293346807}{10349078348235501837504041225} a + \frac{10652900796896462329460478}{71372954125762081637958905}$

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

Class group and class number

$C_{2}\times C_{6}$, 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:  \( 14321935.1855 \) (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

\(\Q(\sqrt{-6095}) \), 7.1.226423307375.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.226423307375.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/29.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
$53$53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$