Properties

Label 14.0.1789940649848551.1
Degree $14$
Signature $[0, 7]$
Discriminant $-1.790\times 10^{15}$
Root discriminant $12.29$
Ramified prime $151$
Class number $1$
Class group trivial
Galois group $D_{7}$ (as 14T2)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 4*x^13 + 6*x^12 - 2*x^11 - 9*x^10 + 19*x^9 - 16*x^8 - 5*x^7 + 25*x^6 - 22*x^5 + 41*x^4 - 60*x^3 + 23*x^2 + 3*x + 9)
 
gp: K = bnfinit(x^14 - 4*x^13 + 6*x^12 - 2*x^11 - 9*x^10 + 19*x^9 - 16*x^8 - 5*x^7 + 25*x^6 - 22*x^5 + 41*x^4 - 60*x^3 + 23*x^2 + 3*x + 9, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 3, 23, -60, 41, -22, 25, -5, -16, 19, -9, -2, 6, -4, 1]);
 

\(x^{14} - 4 x^{13} + 6 x^{12} - 2 x^{11} - 9 x^{10} + 19 x^{9} - 16 x^{8} - 5 x^{7} + 25 x^{6} - 22 x^{5} + 41 x^{4} - 60 x^{3} + 23 x^{2} + 3 x + 9\)  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:  $[0, 7]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-1789940649848551\)\(\medspace = -\,151^{7}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $12.29$
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:  $151$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $14$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{21} a^{12} - \frac{2}{21} a^{11} - \frac{1}{7} a^{10} + \frac{2}{21} a^{9} + \frac{1}{7} a^{8} - \frac{2}{7} a^{7} + \frac{2}{7} a^{6} + \frac{3}{7} a^{5} - \frac{1}{21} a^{4} + \frac{8}{21} a^{3} + \frac{2}{7} a^{2} + \frac{10}{21} a + \frac{2}{7}$, $\frac{1}{37345623} a^{13} + \frac{186566}{37345623} a^{12} + \frac{3276353}{37345623} a^{11} + \frac{4913759}{37345623} a^{10} - \frac{1890624}{12448541} a^{9} + \frac{3073951}{37345623} a^{8} - \frac{4359268}{37345623} a^{7} - \frac{6323249}{37345623} a^{6} - \frac{1001540}{5335089} a^{5} + \frac{7880597}{37345623} a^{4} - \frac{5324514}{12448541} a^{3} - \frac{1237237}{37345623} a^{2} + \frac{12658874}{37345623} a - \frac{1028957}{12448541}$  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:  $6$
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:  \( 70.4788147496 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{7}\cdot 70.4788147496 \cdot 1}{2\sqrt{1789940649848551}}\approx 0.322009258774$

Galois group

$D_7$ (as 14T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 14
The 5 conjugacy class representatives for $D_{7}$
Character table for $D_{7}$

Intermediate fields

\(\Q(\sqrt{-151}) \), 7.1.3442951.1 x7

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

Sibling fields

Degree 7 sibling: 7.1.3442951.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.7.0.1}{7} }^{2}$ ${\href{/padicField/3.2.0.1}{2} }^{7}$ ${\href{/padicField/5.7.0.1}{7} }^{2}$ ${\href{/padicField/7.2.0.1}{2} }^{7}$ ${\href{/padicField/11.7.0.1}{7} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{7}$ ${\href{/padicField/17.7.0.1}{7} }^{2}$ ${\href{/padicField/19.7.0.1}{7} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{7}$ ${\href{/padicField/29.7.0.1}{7} }^{2}$ ${\href{/padicField/31.7.0.1}{7} }^{2}$ ${\href{/padicField/37.7.0.1}{7} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{7}$ ${\href{/padicField/43.7.0.1}{7} }^{2}$ ${\href{/padicField/47.7.0.1}{7} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{7}$ ${\href{/padicField/59.7.0.1}{7} }^{2}$

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
$151$151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$