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Properties

Label 14.14.8902889397...3125.1
Degree $14$
Signature $[14, 0]$
Discriminant $5^{7}\cdot 7^{24}\cdot 29^{6}$
Root discriminant $266.05$
Ramified primes $5, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_7 \wr C_2$ (as 14T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![95554304, 239195712, -19133968, -185001208, -44562560, 30456902, 9524963, -1755568, -641613, 40460, 18424, -322, -231, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 231*x^12 - 322*x^11 + 18424*x^10 + 40460*x^9 - 641613*x^8 - 1755568*x^7 + 9524963*x^6 + 30456902*x^5 - 44562560*x^4 - 185001208*x^3 - 19133968*x^2 + 239195712*x + 95554304)
 
gp: K = bnfinit(x^14 - 231*x^12 - 322*x^11 + 18424*x^10 + 40460*x^9 - 641613*x^8 - 1755568*x^7 + 9524963*x^6 + 30456902*x^5 - 44562560*x^4 - 185001208*x^3 - 19133968*x^2 + 239195712*x + 95554304, 1)
 

Normalized defining polynomial

\( x^{14} - 231 x^{12} - 322 x^{11} + 18424 x^{10} + 40460 x^{9} - 641613 x^{8} - 1755568 x^{7} + 9524963 x^{6} + 30456902 x^{5} - 44562560 x^{4} - 185001208 x^{3} - 19133968 x^{2} + 239195712 x + 95554304 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8902889397738900740239456696953125=5^{7}\cdot 7^{24}\cdot 29^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $266.05$
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:  $5, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{976} a^{11} + \frac{43}{976} a^{10} + \frac{9}{122} a^{9} - \frac{15}{122} a^{8} - \frac{7}{61} a^{7} + \frac{55}{244} a^{6} + \frac{7}{976} a^{5} + \frac{1}{16} a^{4} - \frac{63}{244} a^{3} + \frac{9}{61} a^{2} - \frac{21}{61} a$, $\frac{1}{37088} a^{12} - \frac{1}{4636} a^{11} - \frac{2121}{37088} a^{10} - \frac{169}{4636} a^{9} + \frac{101}{2318} a^{8} + \frac{1727}{9272} a^{7} - \frac{4381}{37088} a^{6} + \frac{250}{1159} a^{5} - \frac{2875}{37088} a^{4} + \frac{4591}{9272} a^{3} - \frac{785}{2318} a^{2} - \frac{2189}{4636} a - \frac{9}{19}$, $\frac{1}{35217623345150664473605807576855992704} a^{13} - \frac{106398440992233568959167470209393}{17608811672575332236802903788427996352} a^{12} + \frac{5401469863349532210144977513163541}{35217623345150664473605807576855992704} a^{11} - \frac{482957063025981184453932096803241595}{8804405836287666118401451894213998176} a^{10} - \frac{432834783549229537619980186646538049}{4402202918143833059200725947106999088} a^{9} - \frac{81837228666773351706434956567806649}{8804405836287666118401451894213998176} a^{8} + \frac{7383178344709043366038284663132901371}{35217623345150664473605807576855992704} a^{7} - \frac{1586007052574398474803440982811894355}{17608811672575332236802903788427996352} a^{6} - \frac{5386562634481001552780717205591446377}{35217623345150664473605807576855992704} a^{5} - \frac{756871052191831403209707049945162597}{4402202918143833059200725947106999088} a^{4} + \frac{1330793327662639872761196963317459657}{4402202918143833059200725947106999088} a^{3} + \frac{2055130497997594620191091865946486099}{4402202918143833059200725947106999088} a^{2} + \frac{228946754846085868917016926510668485}{550275364767979132400090743388374886} a - \frac{1567612120769154523399178611590131}{4510453809573599445902383142527663}$

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:  $13$
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:  \( 24648398601900 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7\times D_7$ (as 14T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 98
The 35 conjugacy class representatives for $C_7 \wr C_2$
Character table for $C_7 \wr C_2$ is not computed

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ R R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ R ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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$5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
7Data not computed
$29$29.7.6.5$x^{7} + 58$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.0.1$x^{7} - x + 3$$1$$7$$0$$C_7$$[\ ]^{7}$