Properties

Label 14.2.415...224.2
Degree $14$
Signature $[2, 6]$
Discriminant $4.156\times 10^{18}$
Root discriminant $21.38$
Ramified primes $2, 317$
Class number $3$
Class group $[3]$
Galois group $\PSL(2,7)$ (as 14T10)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + x^12 + 7*x^11 + 9*x^10 - 5*x^9 + 21*x^8 + 63*x^7 + 32*x^6 + 74*x^5 + 188*x^4 + 70*x^3 - 230*x^2 - 144*x - 22)
 
gp: K = bnfinit(x^14 - x^13 + x^12 + 7*x^11 + 9*x^10 - 5*x^9 + 21*x^8 + 63*x^7 + 32*x^6 + 74*x^5 + 188*x^4 + 70*x^3 - 230*x^2 - 144*x - 22, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-22, -144, -230, 70, 188, 74, 32, 63, 21, -5, 9, 7, 1, -1, 1]);
 

\(x^{14} - x^{13} + x^{12} + 7 x^{11} + 9 x^{10} - 5 x^{9} + 21 x^{8} + 63 x^{7} + 32 x^{6} + 74 x^{5} + 188 x^{4} + 70 x^{3} - 230 x^{2} - 144 x - 22\)  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:  $[2, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(4156382630830772224\)\(\medspace = 2^{12}\cdot 317^{6}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $21.38$
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:  $2, 317$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{9} - \frac{1}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} - \frac{1}{16} a^{8} + \frac{5}{16} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{8}$, $\frac{1}{42045951392} a^{13} + \frac{280105071}{10511487848} a^{12} - \frac{1674539243}{42045951392} a^{11} + \frac{260012967}{2627871962} a^{10} - \frac{8463272311}{42045951392} a^{9} - \frac{1278847713}{2627871962} a^{8} - \frac{7847461091}{42045951392} a^{7} - \frac{136497702}{1313935981} a^{6} - \frac{87970726}{1313935981} a^{5} + \frac{5876552373}{21022975696} a^{4} + \frac{507119023}{21022975696} a^{3} + \frac{154263775}{10511487848} a^{2} + \frac{9430707683}{21022975696} a - \frac{5858881825}{21022975696}$  Toggle raw display

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

Class group and class number

$C_{3}$, which has order $3$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
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:  \( 5502.73317638 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{6}\cdot 5502.73317638 \cdot 3}{2\sqrt{4156382630830772224}}\approx 0.996440003506$

Galois group

$\PSL(2,7)$ (as 14T10):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 168
The 6 conjugacy class representatives for $\PSL(2,7)$
Character table for $\PSL(2,7)$

Intermediate fields

7.3.6431296.2

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

Sibling fields

Degree 7 siblings: 7.3.6431296.2, 7.3.6431296.1
Degree 8 sibling: 8.0.646274503744.1
Degree 21 sibling: 21.5.26730926988131422081122304.1
Degree 24 sibling: data not computed
Degree 28 sibling: data not computed
Degree 42 siblings: data not computed
Arithmetically equvalently sibling: 14.2.4156382630830772224.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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

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$2.7.6.1$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}$
317Data not computed