Properties

Label 14.2.415...224.1
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 - 7*x^12 + 9*x^11 + 25*x^10 + 3*x^9 - 117*x^8 - 73*x^7 + 406*x^6 + 26*x^5 - 480*x^4 - 164*x^3 + 456*x^2 + 204*x - 116)
 
gp: K = bnfinit(x^14 - x^13 - 7*x^12 + 9*x^11 + 25*x^10 + 3*x^9 - 117*x^8 - 73*x^7 + 406*x^6 + 26*x^5 - 480*x^4 - 164*x^3 + 456*x^2 + 204*x - 116, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-116, 204, 456, -164, -480, 26, 406, -73, -117, 3, 25, 9, -7, -1, 1]);
 

\(x^{14} - x^{13} - 7 x^{12} + 9 x^{11} + 25 x^{10} + 3 x^{9} - 117 x^{8} - 73 x^{7} + 406 x^{6} + 26 x^{5} - 480 x^{4} - 164 x^{3} + 456 x^{2} + 204 x - 116\)  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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{1891874804997038} a^{13} + \frac{218102613586431}{1891874804997038} a^{12} + \frac{187073085814668}{945937402498519} a^{11} - \frac{191810521200383}{945937402498519} a^{10} + \frac{165760374023856}{945937402498519} a^{9} + \frac{339970133997857}{1891874804997038} a^{8} + \frac{455553928093765}{945937402498519} a^{7} - \frac{322970410798278}{945937402498519} a^{6} + \frac{835027717449179}{1891874804997038} a^{5} + \frac{35517148568566}{945937402498519} a^{4} + \frac{50546028537219}{945937402498519} a^{3} - \frac{12602477137948}{945937402498519} a^{2} - \frac{28314009263874}{945937402498519} a - \frac{172258960003668}{945937402498519}$  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

7.3.6431296.1

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: 14.2.4156382630830772224.2

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