↑

Properties

Label 14.0.29898464040...7488.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{12}\cdot 3^{12}\cdot 7^{11}\cdot 11^{12}\cdot 19^{12}$
Root discriminant $2087.80$
Ramified primes $2, 3, 7, 11, 19$
Class number Not computed
Class group Not computed
Galois group $F_7$ (as 14T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![413845826258, 40527578, -148601642, 22513666, 112580482, -40530910, -4500440, 1284559, 1337, -721, 329, -119, 35, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 35*x^12 - 119*x^11 + 329*x^10 - 721*x^9 + 1337*x^8 + 1284559*x^7 - 4500440*x^6 - 40530910*x^5 + 112580482*x^4 + 22513666*x^3 - 148601642*x^2 + 40527578*x + 413845826258)
 
gp: K = bnfinit(x^14 - 7*x^13 + 35*x^12 - 119*x^11 + 329*x^10 - 721*x^9 + 1337*x^8 + 1284559*x^7 - 4500440*x^6 - 40530910*x^5 + 112580482*x^4 + 22513666*x^3 - 148601642*x^2 + 40527578*x + 413845826258, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 35 x^{12} - 119 x^{11} + 329 x^{10} - 721 x^{9} + 1337 x^{8} + 1284559 x^{7} - 4500440 x^{6} - 40530910 x^{5} + 112580482 x^{4} + 22513666 x^{3} - 148601642 x^{2} + 40527578 x + 413845826258 \)

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:  \(-29898464040397228799207555080392669309147967488=-\,2^{12}\cdot 3^{12}\cdot 7^{11}\cdot 11^{12}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2087.80$
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:  $2, 3, 7, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{21} a^{6} + \frac{2}{21} a^{5} - \frac{1}{7} a^{4} + \frac{1}{21} a^{3} + \frac{3}{7} a^{2} + \frac{4}{21} a - \frac{2}{7}$, $\frac{1}{21} a^{7} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{2}{21}$, $\frac{1}{1197} a^{8} - \frac{4}{1197} a^{7} + \frac{2}{171} a^{6} - \frac{4}{171} a^{5} + \frac{7}{171} a^{4} - \frac{8}{171} a^{3} + \frac{8}{171} a^{2} + \frac{481}{1197} a + \frac{358}{1197}$, $\frac{1}{1197} a^{9} - \frac{2}{1197} a^{7} + \frac{4}{171} a^{6} - \frac{1}{19} a^{5} + \frac{20}{171} a^{4} - \frac{8}{57} a^{3} - \frac{164}{399} a^{2} - \frac{16}{171} a + \frac{235}{1197}$, $\frac{1}{8379} a^{10} + \frac{1}{8379} a^{9} - \frac{2}{8379} a^{8} + \frac{83}{8379} a^{7} - \frac{5}{1197} a^{6} - \frac{160}{1197} a^{5} - \frac{4}{1197} a^{4} + \frac{237}{931} a^{3} + \frac{3785}{8379} a^{2} + \frac{191}{931} a - \frac{3470}{8379}$, $\frac{1}{8379} a^{11} - \frac{1}{2793} a^{9} + \frac{1}{8379} a^{8} - \frac{181}{8379} a^{7} + \frac{1}{63} a^{6} - \frac{1}{57} a^{5} - \frac{758}{8379} a^{4} - \frac{346}{1197} a^{3} + \frac{811}{8379} a^{2} + \frac{3085}{8379} a + \frac{1727}{8379}$, $\frac{1}{25137} a^{12} + \frac{4}{25137} a^{9} + \frac{1}{2793} a^{8} - \frac{134}{8379} a^{7} + \frac{2}{513} a^{6} - \frac{358}{2793} a^{5} - \frac{118}{1197} a^{4} - \frac{1678}{3591} a^{3} - \frac{167}{931} a^{2} + \frac{1667}{8379} a - \frac{4082}{25137}$, $\frac{1}{1781544021126816802886406569743409708667} a^{13} + \frac{1137187306261040970565591435285229}{1781544021126816802886406569743409708667} a^{12} - \frac{26341033044922481569857752350484548}{593848007042272267628802189914469902889} a^{11} - \frac{20157663048454007228544362640444833}{1781544021126816802886406569743409708667} a^{10} + \frac{23656743263423146131918164736516005}{93765474796148252783495082618074195193} a^{9} - \frac{38770199131669151764716922343073056}{593848007042272267628802189914469902889} a^{8} + \frac{227135736440998490577585947607583910}{93765474796148252783495082618074195193} a^{7} - \frac{15655356354845389936780425275363445275}{1781544021126816802886406569743409708667} a^{6} - \frac{55545298882817551985396517334036109348}{593848007042272267628802189914469902889} a^{5} - \frac{76133404482749969972832508422196401421}{1781544021126816802886406569743409708667} a^{4} + \frac{828775730200243673086395950457303666856}{1781544021126816802886406569743409708667} a^{3} + \frac{50440132336728928632482791807003299101}{197949335680757422542934063304823300963} a^{2} - \frac{433493208491727738081325668966533414216}{1781544021126816802886406569743409708667} a - \frac{287721605340749388208582746030518942743}{1781544021126816802886406569743409708667}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_7$ (as 14T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

\(\Q(\sqrt{-7}) \), 7.1.65354488358705521601472.1

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

Sibling fields

Galois closure: data not computed
Degree 7 sibling: data not computed
Degree 21 sibling: 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R R ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
$3$3.14.12.1$x^{14} - 3 x^{7} + 18$$7$$2$$12$$F_7$$[\ ]_{7}^{6}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$11$11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
$19$19.14.12.1$x^{14} - 19 x^{7} + 722$$7$$2$$12$$F_7$$[\ ]_{7}^{6}$