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Properties

Label 14.0.78670808148...7792.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{12}\cdot 3^{12}\cdot 7^{14}\cdot 127^{7}$
Root discriminant $366.43$
Ramified primes $2, 3, 7, 127$
Class number $560$ (GRH)
Class group $[2, 2, 2, 70]$ (GRH)
Galois group $F_7 \times C_2$ (as 14T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![34364905256, -7525686616, 8218335496, -1444372664, 815970232, -114741256, 43689016, -4803857, 1365287, -112021, 24899, -1379, 245, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 245*x^12 - 1379*x^11 + 24899*x^10 - 112021*x^9 + 1365287*x^8 - 4803857*x^7 + 43689016*x^6 - 114741256*x^5 + 815970232*x^4 - 1444372664*x^3 + 8218335496*x^2 - 7525686616*x + 34364905256)
 
gp: K = bnfinit(x^14 - 7*x^13 + 245*x^12 - 1379*x^11 + 24899*x^10 - 112021*x^9 + 1365287*x^8 - 4803857*x^7 + 43689016*x^6 - 114741256*x^5 + 815970232*x^4 - 1444372664*x^3 + 8218335496*x^2 - 7525686616*x + 34364905256, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 245 x^{12} - 1379 x^{11} + 24899 x^{10} - 112021 x^{9} + 1365287 x^{8} - 4803857 x^{7} + 43689016 x^{6} - 114741256 x^{5} + 815970232 x^{4} - 1444372664 x^{3} + 8218335496 x^{2} - 7525686616 x + 34364905256 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-786708081482146423456402960489377792=-\,2^{12}\cdot 3^{12}\cdot 7^{14}\cdot 127^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $366.43$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 7, 127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{5}$, $\frac{1}{605254637567760787880241928331492740103560444} a^{13} - \frac{11767685003640801386326240951049789831090012}{151313659391940196970060482082873185025890111} a^{12} - \frac{17007536467680674015515862696448218742057159}{302627318783880393940120964165746370051780222} a^{11} - \frac{12476314738282603619345210727419928401544745}{302627318783880393940120964165746370051780222} a^{10} + \frac{59789921489342223055980975340967778001653607}{302627318783880393940120964165746370051780222} a^{9} - \frac{39191004880064744702493501916191373059542543}{302627318783880393940120964165746370051780222} a^{8} - \frac{295054193898911567756663317841157970607553}{151313659391940196970060482082873185025890111} a^{7} + \frac{52253658827931265050019764965035698743757421}{302627318783880393940120964165746370051780222} a^{6} - \frac{70431167755261453235293295490132574666332191}{605254637567760787880241928331492740103560444} a^{5} - \frac{10918297611978257710195233460659233108545625}{302627318783880393940120964165746370051780222} a^{4} + \frac{19205267745458050837696045334419592289251287}{302627318783880393940120964165746370051780222} a^{3} - \frac{62147635882679377174448013045888089687243388}{151313659391940196970060482082873185025890111} a^{2} - \frac{19331362799975480511488664573726553141709488}{151313659391940196970060482082873185025890111} a + \frac{48233242153216903820637168844655342763555059}{151313659391940196970060482082873185025890111}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{70}$, which has order $560$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $6$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 11589105043.682163 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_7$ (as 14T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 84
The 14 conjugacy class representatives for $F_7 \times C_2$
Character table for $F_7 \times C_2$

Intermediate fields

\(\Q(\sqrt{-127}) \), 7.1.38423222208.5

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 sibling: 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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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

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];
 
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]])
 

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}$
$3$3.14.12.1$x^{14} - 3 x^{7} + 18$$7$$2$$12$$F_7$$[\ ]_{7}^{6}$
$7$7.14.14.21$x^{14} + 28 x^{12} + 42 x^{11} + 42 x^{9} + 21 x^{8} + 29 x^{7} + 21 x^{6} + 35 x^{5} + 7 x^{4} + 14 x^{3} + 28 x^{2} + 42 x + 45$$7$$2$$14$$F_7 \times C_2$$[7/6]_{6}^{2}$
$127$127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.2$x^{2} + 1143$$2$$1$$1$$C_2$$[\ ]_{2}$