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Properties

Label 14.0.25230488364...0000.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{12}\cdot 3^{12}\cdot 5^{12}\cdot 7^{15}$
Root discriminant $148.45$
Ramified primes $2, 3, 5, 7$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $F_7$ (as 14T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1455728, 75152, -276080, 40096, 212632, -78484, -5726, 355, 1337, -721, 329, -119, 35, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 35*x^12 - 119*x^11 + 329*x^10 - 721*x^9 + 1337*x^8 + 355*x^7 - 5726*x^6 - 78484*x^5 + 212632*x^4 + 40096*x^3 - 276080*x^2 + 75152*x + 1455728)
 
gp: K = bnfinit(x^14 - 7*x^13 + 35*x^12 - 119*x^11 + 329*x^10 - 721*x^9 + 1337*x^8 + 355*x^7 - 5726*x^6 - 78484*x^5 + 212632*x^4 + 40096*x^3 - 276080*x^2 + 75152*x + 1455728, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 35 x^{12} - 119 x^{11} + 329 x^{10} - 721 x^{9} + 1337 x^{8} + 355 x^{7} - 5726 x^{6} - 78484 x^{5} + 212632 x^{4} + 40096 x^{3} - 276080 x^{2} + 75152 x + 1455728 \)

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:  \(-2523048836405617863000000000000=-\,2^{12}\cdot 3^{12}\cdot 5^{12}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $148.45$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{14} a^{7} - \frac{1}{2} a^{3} + \frac{3}{7}$, $\frac{1}{140} a^{8} - \frac{1}{35} a^{7} + \frac{1}{10} a^{6} - \frac{1}{5} a^{5} - \frac{3}{20} a^{4} - \frac{2}{5} a^{3} - \frac{1}{10} a^{2} + \frac{12}{35} a - \frac{6}{35}$, $\frac{1}{140} a^{9} - \frac{1}{70} a^{7} + \frac{1}{5} a^{6} + \frac{1}{20} a^{5} + \frac{3}{10} a^{3} - \frac{2}{35} a^{2} + \frac{1}{5} a + \frac{11}{35}$, $\frac{1}{140} a^{10} - \frac{1}{4} a^{6} + \frac{1}{10} a^{5} - \frac{5}{14} a^{3} - \frac{1}{2} a^{2} - \frac{1}{5}$, $\frac{1}{980} a^{11} + \frac{3}{980} a^{10} - \frac{1}{980} a^{9} + \frac{3}{980} a^{8} + \frac{3}{196} a^{7} - \frac{11}{140} a^{6} + \frac{3}{140} a^{5} - \frac{183}{980} a^{4} + \frac{17}{49} a^{3} - \frac{157}{490} a^{2} - \frac{13}{245} a + \frac{22}{49}$, $\frac{1}{1960} a^{12} - \frac{3}{1960} a^{10} + \frac{3}{980} a^{9} - \frac{1}{1960} a^{8} - \frac{3}{245} a^{7} + \frac{57}{280} a^{6} - \frac{221}{980} a^{5} + \frac{1}{35} a^{4} + \frac{167}{490} a^{3} - \frac{243}{490} a^{2} - \frac{18}{49} a - \frac{116}{245}$, $\frac{1}{18505507015664006784040} a^{13} - \frac{3384643915717365151}{18505507015664006784040} a^{12} + \frac{8197959353850922219}{18505507015664006784040} a^{11} + \frac{9713245663503004775}{3701101403132801356808} a^{10} - \frac{22720868507802259309}{18505507015664006784040} a^{9} - \frac{33985216290722843299}{18505507015664006784040} a^{8} + \frac{69616162499235730621}{18505507015664006784040} a^{7} + \frac{437288523842181383077}{18505507015664006784040} a^{6} - \frac{585878468634889964071}{4626376753916001696010} a^{5} - \frac{2089541699216947091321}{9252753507832003392020} a^{4} + \frac{181519229046313272404}{2313188376958000848005} a^{3} + \frac{607455728931398595473}{4626376753916001696010} a^{2} + \frac{223905874224576490204}{2313188376958000848005} a - \frac{975020100803981193638}{2313188376958000848005}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( 1187793284.646849 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_7$ (as 14T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

\(\Q(\sqrt{-7}) \), 7.1.600362847000000.17

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

Sibling fields

Galois closure: data not computed
Degree 7 sibling: data not computed
Degree 21 sibling: 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 R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$$[\ ]_{7}^{6}$
$5$5.14.12.1$x^{14} - 5 x^{7} + 50$$7$$2$$12$$F_7$$[\ ]_{7}^{6}$
$7$7.14.15.5$x^{14} - 21 x^{13} + 7 x^{12} + 14 x^{11} - 21 x^{9} + 7 x^{7} + 14 x^{6} - 21 x^{4} + 14 x^{3} + 7 x^{2} + 14$$14$$1$$15$$F_7$$[7/6]_{6}$