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Properties

Label 14.14.2267090493...1057.2
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![-26099424, 118708648, -168908516, 27685502, 118055451, -73124614, -3173277, 7571477, -545963, -244901, 27489, 2458, -333, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 6*x^13 - 333*x^12 + 2458*x^11 + 27489*x^10 - 244901*x^9 - 545963*x^8 + 7571477*x^7 - 3173277*x^6 - 73124614*x^5 + 118055451*x^4 + 27685502*x^3 - 168908516*x^2 + 118708648*x - 26099424)
 
gp: K = bnfinit(x^14 - 6*x^13 - 333*x^12 + 2458*x^11 + 27489*x^10 - 244901*x^9 - 545963*x^8 + 7571477*x^7 - 3173277*x^6 - 73124614*x^5 + 118055451*x^4 + 27685502*x^3 - 168908516*x^2 + 118708648*x - 26099424, 1)
 

Normalized defining polynomial

\( x^{14} - 6 x^{13} - 333 x^{12} + 2458 x^{11} + 27489 x^{10} - 244901 x^{9} - 545963 x^{8} + 7571477 x^{7} - 3173277 x^{6} - 73124614 x^{5} + 118055451 x^{4} + 27685502 x^{3} - 168908516 x^{2} + 118708648 x - 26099424 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{24} a^{10} - \frac{5}{12} a^{8} - \frac{1}{12} a^{7} + \frac{1}{8} a^{6} - \frac{3}{8} a^{5} + \frac{1}{3} a^{4} + \frac{5}{12} a^{3} - \frac{5}{24} a^{2} - \frac{1}{12} a$, $\frac{1}{10392} a^{11} - \frac{83}{10392} a^{10} + \frac{481}{5196} a^{9} + \frac{83}{866} a^{8} - \frac{2231}{10392} a^{7} - \frac{571}{1732} a^{6} - \frac{2857}{10392} a^{5} - \frac{439}{1732} a^{4} + \frac{3821}{10392} a^{3} - \frac{3691}{10392} a^{2} - \frac{2359}{5196} a - \frac{77}{433}$, $\frac{1}{498816} a^{12} + \frac{1}{166272} a^{11} - \frac{193}{15588} a^{10} + \frac{2635}{31176} a^{9} - \frac{145199}{498816} a^{8} + \frac{60293}{124704} a^{7} + \frac{180539}{498816} a^{6} + \frac{13057}{31176} a^{5} + \frac{224153}{498816} a^{4} + \frac{21091}{55424} a^{3} + \frac{4547}{31176} a^{2} - \frac{18763}{124704} a - \frac{1681}{3464}$, $\frac{1}{3380457285425176177076386472723328} a^{13} + \frac{1369428927279135524988036299}{3380457285425176177076386472723328} a^{12} + \frac{11028729120654035701225301591}{422557160678147022134548309090416} a^{11} + \frac{85332953159236333474372150309}{7825132605150870780269413131304} a^{10} + \frac{95428031246941514313291638555}{1126819095141725392358795490907776} a^{9} + \frac{39632555408310815169804471702775}{845114321356294044269096618180832} a^{8} + \frac{140108988306000677801135696774083}{375606365047241797452931830302592} a^{7} + \frac{154578755293781624266099258195309}{422557160678147022134548309090416} a^{6} - \frac{963202203035475636016575019593767}{3380457285425176177076386472723328} a^{5} - \frac{12729519947163290332912313355773}{3380457285425176177076386472723328} a^{4} - \frac{30642447430674124068490949944967}{422557160678147022134548309090416} a^{3} + \frac{34177953602742588977168784591863}{281704773785431348089698872726944} a^{2} - \frac{34960767783268281585093880972007}{211278580339073511067274154545208} a + \frac{1342010078435103016590638900077}{2934424726931576542601029924239}$

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:  \( 3505494749950000000 \) (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 ${\href{/LocalNumberField/2.7.0.1}{7} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/3.7.0.1}{7} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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