↑

Properties

Label 14.2.18709104627...5625.2
Degree $14$
Signature $[2, 6]$
Discriminant $5^{10}\cdot 7^{24}$
Root discriminant $88.72$
Ramified primes $5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_7:D_7.C_2$ (as 14T12)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3344875, -396375, -1769950, 237825, 369145, -122360, -64680, 10485, 5614, -203, -56, -154, 14, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 14*x^12 - 154*x^11 - 56*x^10 - 203*x^9 + 5614*x^8 + 10485*x^7 - 64680*x^6 - 122360*x^5 + 369145*x^4 + 237825*x^3 - 1769950*x^2 - 396375*x + 3344875)
 
gp: K = bnfinit(x^14 + 14*x^12 - 154*x^11 - 56*x^10 - 203*x^9 + 5614*x^8 + 10485*x^7 - 64680*x^6 - 122360*x^5 + 369145*x^4 + 237825*x^3 - 1769950*x^2 - 396375*x + 3344875, 1)
 

Normalized defining polynomial

\( x^{14} + 14 x^{12} - 154 x^{11} - 56 x^{10} - 203 x^{9} + 5614 x^{8} + 10485 x^{7} - 64680 x^{6} - 122360 x^{5} + 369145 x^{4} + 237825 x^{3} - 1769950 x^{2} - 396375 x + 3344875 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1870910462700843890634765625=5^{10}\cdot 7^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.72$
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:  $5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} + \frac{3}{7}$, $\frac{1}{7} a^{8} + \frac{3}{7} a$, $\frac{1}{35} a^{9} + \frac{2}{35} a^{8} - \frac{2}{35} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{7} a^{2} - \frac{3}{7} a + \frac{3}{7}$, $\frac{1}{70} a^{10} - \frac{1}{70} a^{8} - \frac{2}{35} a^{7} + \frac{1}{5} a^{6} + \frac{1}{10} a^{5} - \frac{3}{10} a^{4} + \frac{1}{7} a^{3} - \frac{1}{2} a^{2} + \frac{5}{14} a - \frac{1}{14}$, $\frac{1}{70} a^{11} - \frac{1}{70} a^{9} - \frac{2}{35} a^{8} + \frac{2}{35} a^{7} + \frac{1}{10} a^{6} - \frac{3}{10} a^{5} + \frac{1}{7} a^{4} - \frac{1}{2} a^{3} + \frac{5}{14} a^{2} - \frac{1}{14} a - \frac{3}{7}$, $\frac{1}{700} a^{12} - \frac{1}{140} a^{11} + \frac{1}{175} a^{10} + \frac{1}{700} a^{9} - \frac{1}{700} a^{8} - \frac{33}{700} a^{7} + \frac{1}{50} a^{6} - \frac{13}{70} a^{5} - \frac{6}{35} a^{4} - \frac{31}{70} a^{3} - \frac{61}{140} a^{2} + \frac{2}{7} a + \frac{5}{28}$, $\frac{1}{383971722949510506630436368500} a^{13} - \frac{47266846339868695439362021}{76794344589902101326087273700} a^{12} + \frac{112622493786185272776478771}{27426551639250750473602597750} a^{11} - \frac{1828535177838464708281591449}{383971722949510506630436368500} a^{10} + \frac{2627075138598044584237169509}{383971722949510506630436368500} a^{9} - \frac{38504885095642818176290719}{772578919415514097847960500} a^{8} + \frac{10859739974132099279466256687}{191985861474755253315218184250} a^{7} + \frac{632348870280406873315846108}{3839717229495105066304363685} a^{6} + \frac{8314503197638880005856965837}{19198586147475525331521818425} a^{5} - \frac{1083731135599637574146873454}{2742655163925075047360259775} a^{4} - \frac{29912930876889011914446820571}{76794344589902101326087273700} a^{3} + \frac{1243925733644023738743465868}{3839717229495105066304363685} a^{2} - \frac{62973217690796976560748937}{2194124131140060037888207820} a - \frac{177323490117219184825269983}{1535886891798042026521745474}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $7$
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:  \( 352671602.612 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7:D_7.C_2$ (as 14T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 196
The 16 conjugacy class representatives for $C_7:D_7.C_2$
Character table for $C_7:D_7.C_2$

Intermediate fields

\(\Q(\sqrt{5}) \)

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

Sibling fields

Degree 14 siblings: data not computed
Degree 28 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
7Data not computed