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Properties

Label 14.14.2267090493...1057.1
Degree $14$
Signature $[14, 0]$
Discriminant $577^{7}\cdot 1009^{7}$
Root discriminant $763.02$
Ramified primes $577, 1009$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_7^2$ (as 14T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-963639599, 485583417, 701452818, -272636914, -195969927, 48357153, 25343855, -2857477, -1444514, 56049, 36188, -72, -349, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 349*x^12 - 72*x^11 + 36188*x^10 + 56049*x^9 - 1444514*x^8 - 2857477*x^7 + 25343855*x^6 + 48357153*x^5 - 195969927*x^4 - 272636914*x^3 + 701452818*x^2 + 485583417*x - 963639599)
 
gp: K = bnfinit(x^14 - 2*x^13 - 349*x^12 - 72*x^11 + 36188*x^10 + 56049*x^9 - 1444514*x^8 - 2857477*x^7 + 25343855*x^6 + 48357153*x^5 - 195969927*x^4 - 272636914*x^3 + 701452818*x^2 + 485583417*x - 963639599, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} - 349 x^{12} - 72 x^{11} + 36188 x^{10} + 56049 x^{9} - 1444514 x^{8} - 2857477 x^{7} + 25343855 x^{6} + 48357153 x^{5} - 195969927 x^{4} - 272636914 x^{3} + 701452818 x^{2} + 485583417 x - 963639599 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(22670904930664086126843692355192338431057=577^{7}\cdot 1009^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $763.02$
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:  $577, 1009$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{14315176157233272511096972840470722680407897455149} a^{13} + \frac{3342651976528067253980471540101392399577569993849}{14315176157233272511096972840470722680407897455149} a^{12} - \frac{2577486122383116264443297636032297429250292511842}{14315176157233272511096972840470722680407897455149} a^{11} - \frac{2518642741452995151117724074356965038684673637305}{14315176157233272511096972840470722680407897455149} a^{10} - \frac{3134578400316434010723905012309032011519780612002}{14315176157233272511096972840470722680407897455149} a^{9} + \frac{5227735427997405442316170342877248044715152137675}{14315176157233272511096972840470722680407897455149} a^{8} - \frac{48090366565264799307094345161316351804358073522}{14315176157233272511096972840470722680407897455149} a^{7} - \frac{5553089319741705085622626522697908406303755166954}{14315176157233272511096972840470722680407897455149} a^{6} + \frac{5795694620705035979673638102768089890817607479305}{14315176157233272511096972840470722680407897455149} a^{5} + \frac{2396717225323961120624132901425908843226698103795}{14315176157233272511096972840470722680407897455149} a^{4} - \frac{633469471312783450568182153498341540434320338010}{14315176157233272511096972840470722680407897455149} a^{3} - \frac{2682768233641440556693457342006319870115185710106}{14315176157233272511096972840470722680407897455149} a^{2} + \frac{4176028591855028484709200354036673821856209344308}{14315176157233272511096972840470722680407897455149} a - \frac{348132401249015899140336461017002269647231580856}{14315176157233272511096972840470722680407897455149}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $13$
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:  \( 1734662984620000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_7^2$ (as 14T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 196
The 25 conjugacy class representatives for $D_7^2$
Character table for $D_7^2$ is not computed

Intermediate fields

\(\Q(\sqrt{582193}) \)

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

Sibling fields

Degree 14 siblings: data not computed
Degree 28 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
577Data not computed
1009Data not computed