↑

Properties

Label 14.0.19118989751...4375.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,5^{7}\cdot 1579^{7}$
Root discriminant $88.85$
Ramified primes $5, 1579$
Class number $16$ (GRH)
Class group $[16]$ (GRH)
Galois group $D_{7}$ (as 14T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1476225, 3553875, 4799250, 4052160, 2442591, 1019523, 306881, 47705, 1448, -2901, -192, -120, 31, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 31*x^12 - 120*x^11 - 192*x^10 - 2901*x^9 + 1448*x^8 + 47705*x^7 + 306881*x^6 + 1019523*x^5 + 2442591*x^4 + 4052160*x^3 + 4799250*x^2 + 3553875*x + 1476225)
 
gp: K = bnfinit(x^14 - 2*x^13 + 31*x^12 - 120*x^11 - 192*x^10 - 2901*x^9 + 1448*x^8 + 47705*x^7 + 306881*x^6 + 1019523*x^5 + 2442591*x^4 + 4052160*x^3 + 4799250*x^2 + 3553875*x + 1476225, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} + 31 x^{12} - 120 x^{11} - 192 x^{10} - 2901 x^{9} + 1448 x^{8} + 47705 x^{7} + 306881 x^{6} + 1019523 x^{5} + 2442591 x^{4} + 4052160 x^{3} + 4799250 x^{2} + 3553875 x + 1476225 \)

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:  \(-1911898975188880445832734375=-\,5^{7}\cdot 1579^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.85$
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:  $5, 1579$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{27} a^{8} - \frac{1}{27} a^{7} + \frac{1}{27} a^{6} - \frac{4}{27} a^{5} + \frac{1}{27} a^{4} - \frac{4}{27} a^{3} - \frac{1}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{81} a^{9} - \frac{1}{27} a^{7} + \frac{1}{27} a^{5} - \frac{10}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{405} a^{10} + \frac{1}{27} a^{7} - \frac{4}{135} a^{6} + \frac{4}{27} a^{5} - \frac{5}{81} a^{4} + \frac{4}{27} a^{3} + \frac{4}{45} a^{2} - \frac{1}{3} a$, $\frac{1}{3645} a^{11} - \frac{1}{3645} a^{10} + \frac{1}{81} a^{8} + \frac{41}{1215} a^{7} - \frac{26}{1215} a^{6} - \frac{59}{729} a^{5} - \frac{13}{729} a^{4} - \frac{26}{405} a^{3} + \frac{11}{405} a^{2} + \frac{4}{9} a$, $\frac{1}{10935} a^{12} + \frac{1}{10935} a^{11} - \frac{2}{10935} a^{10} + \frac{1}{243} a^{9} - \frac{64}{3645} a^{8} - \frac{79}{3645} a^{7} - \frac{46}{10935} a^{6} - \frac{212}{2187} a^{5} - \frac{1174}{10935} a^{4} - \frac{86}{1215} a^{3} + \frac{292}{1215} a^{2} + \frac{11}{27} a$, $\frac{1}{8085190912452081014445} a^{13} + \frac{204603386467484134}{8085190912452081014445} a^{12} + \frac{628424565846636157}{8085190912452081014445} a^{11} + \frac{1147703958045628}{1292596468817279139} a^{10} + \frac{5591447461709337086}{2695063637484027004815} a^{9} + \frac{49821099612994680329}{2695063637484027004815} a^{8} - \frac{382112100293286819889}{8085190912452081014445} a^{7} + \frac{40999483004364483610}{1617038182490416202889} a^{6} + \frac{620061680411935392191}{8085190912452081014445} a^{5} + \frac{144603930135608206906}{898354545828009001605} a^{4} - \frac{56854975030728060542}{898354545828009001605} a^{3} + \frac{5386257617599940303}{19963434351733533369} a^{2} + \frac{192111643641180625}{2218159372414837041} a - \frac{9847022874587309}{246462152490537449}$

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

Class group and class number

$C_{16}$, which has order $16$ (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:  \( 2301324546.09 \) (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

\(\Q(\sqrt{-7895}) \), 7.1.492103442375.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.492103442375.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.1.0.1}{1} }^{14}$ R ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}$

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
$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}$
1579Data not computed