Properties

Label 14.2.749...313.1
Degree $14$
Signature $[2, 6]$
Discriminant $7.490\times 10^{18}$
Root discriminant $22.29$
Ramified primes $7, 71$
Class number $1$
Class group trivial
Galois group $D_{14}$ (as 14T3)

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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 - 4*x^12 + 32*x^11 - 44*x^10 + 76*x^9 + 222*x^8 - 1472*x^7 + 483*x^6 + 4460*x^5 - 2122*x^4 - 5110*x^3 + 564*x^2 + 2882*x - 1573)
 
gp: K = bnfinit(x^14 - 4*x^13 - 4*x^12 + 32*x^11 - 44*x^10 + 76*x^9 + 222*x^8 - 1472*x^7 + 483*x^6 + 4460*x^5 - 2122*x^4 - 5110*x^3 + 564*x^2 + 2882*x - 1573, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1573, 2882, 564, -5110, -2122, 4460, 483, -1472, 222, 76, -44, 32, -4, -4, 1]);
 

\(x^{14} - 4 x^{13} - 4 x^{12} + 32 x^{11} - 44 x^{10} + 76 x^{9} + 222 x^{8} - 1472 x^{7} + 483 x^{6} + 4460 x^{5} - 2122 x^{4} - 5110 x^{3} + 564 x^{2} + 2882 x - 1573\)  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:  \(7490222540601799313\)\(\medspace = 7^{7}\cdot 71^{7}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $22.29$
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:  $7, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{142} a^{12} + \frac{6}{71} a^{11} - \frac{9}{71} a^{10} - \frac{15}{71} a^{9} - \frac{11}{142} a^{8} - \frac{13}{71} a^{7} - \frac{29}{71} a^{6} + \frac{45}{142} a^{5} + \frac{8}{71} a^{4} - \frac{5}{71} a^{3} - \frac{20}{71} a^{2} + \frac{1}{71} a + \frac{16}{71}$, $\frac{1}{103056253007828616386} a^{13} + \frac{214549857177268675}{103056253007828616386} a^{12} + \frac{12920841392352523505}{103056253007828616386} a^{11} - \frac{7943724388688949411}{103056253007828616386} a^{10} - \frac{552979833739471872}{4684375136719482563} a^{9} + \frac{11997350057896964524}{51528126503914308193} a^{8} + \frac{177483889730225421}{103056253007828616386} a^{7} - \frac{7710994280801093160}{51528126503914308193} a^{6} - \frac{18722545618869873930}{51528126503914308193} a^{5} - \frac{9719910861316050321}{51528126503914308193} a^{4} - \frac{30309764140778403005}{103056253007828616386} a^{3} - \frac{13809648786982047445}{103056253007828616386} a^{2} + \frac{17939323156338701153}{103056253007828616386} a + \frac{2163008170286822630}{4684375136719482563}$  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:  $7$
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:  \( 6632.36500949 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{6}\cdot 6632.36500949 \cdot 1}{2\sqrt{7490222540601799313}}\approx 0.298215535804$

Galois group

$D_{14}$ (as 14T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 28
The 10 conjugacy class representatives for $D_{14}$
Character table for $D_{14}$

Intermediate fields

\(\Q(\sqrt{497}) \), 7.1.357911.1

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

Sibling fields

Galois closure: Deg 28
Degree 14 sibling: 14.0.105496092121152103.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.7.0.1}{7} }^{2}$ ${\href{/padicField/3.14.0.1}{14} }$ ${\href{/padicField/5.14.0.1}{14} }$ R ${\href{/padicField/11.2.0.1}{2} }^{7}$ ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.14.0.1}{14} }$ ${\href{/padicField/23.2.0.1}{2} }^{7}$ ${\href{/padicField/29.7.0.1}{7} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.7.0.1}{7} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.7.0.1}{7} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{7}$ ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$71$71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.497.2t1.a.a$1$ $ 7 \cdot 71 $ \(\Q(\sqrt{497}) \) $C_2$ (as 2T1) $1$ $1$
1.71.2t1.a.a$1$ $ 71 $ \(\Q(\sqrt{-71}) \) $C_2$ (as 2T1) $1$ $-1$
1.7.2t1.a.a$1$ $ 7 $ \(\Q(\sqrt{-7}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.3479.14t3.a.b$2$ $ 7^{2} \cdot 71 $ 14.2.7490222540601799313.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.71.7t2.a.c$2$ $ 71 $ 7.1.357911.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.71.7t2.a.b$2$ $ 71 $ 7.1.357911.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.71.7t2.a.a$2$ $ 71 $ 7.1.357911.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.3479.14t3.a.c$2$ $ 7^{2} \cdot 71 $ 14.2.7490222540601799313.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.3479.14t3.a.a$2$ $ 7^{2} \cdot 71 $ 14.2.7490222540601799313.1 $D_{14}$ (as 14T3) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.