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Properties

Label 14.0.25077284169...6128.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{12}\cdot 3^{12}\cdot 7^{11}\cdot 17^{12}$
Root discriminant $243.03$
Ramified primes $2, 3, 7, 17$
Class number $1$ (GRH)
Class group Trivial (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![10696256, 205184, -752864, 112336, 573832, -208516, -20174, 4483, 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 + 4483*x^7 - 20174*x^6 - 208516*x^5 + 573832*x^4 + 112336*x^3 - 752864*x^2 + 205184*x + 10696256)
 
gp: K = bnfinit(x^14 - 7*x^13 + 35*x^12 - 119*x^11 + 329*x^10 - 721*x^9 + 1337*x^8 + 4483*x^7 - 20174*x^6 - 208516*x^5 + 573832*x^4 + 112336*x^3 - 752864*x^2 + 205184*x + 10696256, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 35 x^{12} - 119 x^{11} + 329 x^{10} - 721 x^{9} + 1337 x^{8} + 4483 x^{7} - 20174 x^{6} - 208516 x^{5} + 573832 x^{4} + 112336 x^{3} - 752864 x^{2} + 205184 x + 10696256 \)

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:  \(-2507728416978832844027390843056128=-\,2^{12}\cdot 3^{12}\cdot 7^{11}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $243.03$
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:  $2, 3, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{28} a^{6} - \frac{5}{28} a^{5} - \frac{3}{28} a^{4} + \frac{1}{28} a^{3} + \frac{1}{14} a^{2} + \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{28} a^{7} + \frac{1}{4} a^{3} + \frac{3}{7}$, $\frac{1}{56} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{2}{7} a$, $\frac{1}{56} a^{9} - \frac{1}{8} a^{5} + \frac{1}{4} a^{3} - \frac{2}{7} a^{2}$, $\frac{1}{784} a^{10} + \frac{1}{784} a^{9} - \frac{1}{392} a^{8} + \frac{3}{392} a^{7} - \frac{1}{112} a^{6} + \frac{23}{112} a^{5} - \frac{1}{7} a^{4} + \frac{19}{98} a^{3} + \frac{33}{98} a^{2} - \frac{5}{49} a + \frac{15}{49}$, $\frac{1}{784} a^{11} - \frac{3}{784} a^{9} - \frac{3}{392} a^{8} - \frac{13}{784} a^{7} - \frac{3}{112} a^{5} - \frac{57}{392} a^{4} - \frac{9}{28} a^{3} + \frac{13}{98} a^{2} - \frac{8}{49} a - \frac{1}{49}$, $\frac{1}{1568} a^{12} - \frac{1}{1568} a^{10} + \frac{5}{784} a^{9} + \frac{11}{1568} a^{8} + \frac{3}{392} a^{7} - \frac{1}{224} a^{6} + \frac{181}{784} a^{5} + \frac{1}{56} a^{4} - \frac{17}{49} a^{3} - \frac{5}{49} a^{2} - \frac{16}{49} a + \frac{22}{49}$, $\frac{1}{828365542850185721696224} a^{13} + \frac{6255158098967338919}{25886423214068303803007} a^{12} - \frac{5098841514603254279}{63720426373091209361248} a^{11} + \frac{9053013843404975547}{414182771425092860848112} a^{10} + \frac{309749869106335032759}{118337934692883674528032} a^{9} - \frac{383972888886500382075}{207091385712546430424056} a^{8} - \frac{12353313277205440570509}{828365542850185721696224} a^{7} + \frac{1565440354183262073383}{414182771425092860848112} a^{6} - \frac{3705664974379814430225}{31860213186545604680624} a^{5} - \frac{10243113540128950549763}{103545692856273215212028} a^{4} + \frac{925883339603441690224}{1991263324159100292539} a^{3} + \frac{141690205141302981736}{3698060459152614829001} a^{2} + \frac{8998217173912796971578}{25886423214068303803007} a - \frac{9541230730754641943201}{25886423214068303803007}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 335473003203.79584 \) (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

\(\Q(\sqrt{-7}) \), 7.1.18927411780570048.3

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$$[\ ]_{7}^{6}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$17$17.14.12.1$x^{14} - 17 x^{7} + 867$$7$$2$$12$$F_7$$[\ ]_{7}^{6}$