Properties

Label 14.12.341...136.1
Degree $14$
Signature $[12, 1]$
Discriminant $-3.414\times 10^{19}$
Root discriminant $24.84$
Ramified primes $2, 19, 809$
Class number $1$
Class group trivial
Galois group 14T51

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Show commands: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 6*x^12 + 53*x^11 - 98*x^10 - 127*x^9 + 309*x^8 + 174*x^7 - 406*x^6 - 232*x^5 + 238*x^4 + 226*x^3 - 48*x^2 - 86*x - 2)
 
gp: K = bnfinit(x^14 - 7*x^13 + 6*x^12 + 53*x^11 - 98*x^10 - 127*x^9 + 309*x^8 + 174*x^7 - 406*x^6 - 232*x^5 + 238*x^4 + 226*x^3 - 48*x^2 - 86*x - 2, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, -86, -48, 226, 238, -232, -406, 174, 309, -127, -98, 53, 6, -7, 1]);
 

\(x^{14} - 7 x^{13} + 6 x^{12} + 53 x^{11} - 98 x^{10} - 127 x^{9} + 309 x^{8} + 174 x^{7} - 406 x^{6} - 232 x^{5} + 238 x^{4} + 226 x^{3} - 48 x^{2} - 86 x - 2\)  Toggle raw display

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

Invariants

Degree:  $14$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[12, 1]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-34135604022671835136\)\(\medspace = -\,2^{22}\cdot 19\cdot 809^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $24.84$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 19, 809$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{234761} a^{13} - \frac{22817}{234761} a^{12} - \frac{407}{10207} a^{11} - \frac{108047}{234761} a^{10} + \frac{30994}{234761} a^{9} - \frac{107896}{234761} a^{8} + \frac{108506}{234761} a^{7} + \frac{63537}{234761} a^{6} - \frac{99723}{234761} a^{5} + \frac{1391}{3979} a^{4} - \frac{9438}{234761} a^{3} + \frac{5169}{234761} a^{2} - \frac{54916}{234761} a - \frac{50822}{234761}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $12$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 183002.663841 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{12}\cdot(2\pi)^{1}\cdot 183002.663841 \cdot 1}{2\sqrt{34135604022671835136}}\approx 0.403054186175$

Galois group

14T51:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 21504
The 48 conjugacy class representatives for [2^7]L(7)=2wrL(7)
Character table for [2^7]L(7)=2wrL(7) is not computed

Intermediate fields

7.7.670188544.2

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

Sibling fields

Degree 14 sibling: data not computed
Degree 28 siblings: data not computed
Degree 42 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 R ${\href{/padicField/3.14.0.1}{14} }$ ${\href{/padicField/5.7.0.1}{7} }^{2}$ ${\href{/padicField/7.7.0.1}{7} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.14.0.1}{14} }$ ${\href{/padicField/17.7.0.1}{7} }^{2}$ R ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.14.0.1}{14} }$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.8.18.65$x^{8} + 6 x^{4} + 4 x^{3} + 6$$8$$1$$18$$S_4\times C_2$$[2, 8/3, 8/3]_{3}^{2}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
$809$Data not computed