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Properties

Label 14.8.11365531860...8736.1
Degree $14$
Signature $[8, 3]$
Discriminant $-\,2^{27}\cdot 7^{12}\cdot 409^{5}\cdot 3767^{3}$
Root discriminant $1009.18$
Ramified primes $2, 7, 409, 3767$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 14T45

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49980715777967, -19410042679742, 366359666251, 536849629536, -59621810879, 2822730482, 393809153, -96986796, 2905773, -101186, -50815, 2684, -29, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 - 29*x^12 + 2684*x^11 - 50815*x^10 - 101186*x^9 + 2905773*x^8 - 96986796*x^7 + 393809153*x^6 + 2822730482*x^5 - 59621810879*x^4 + 536849629536*x^3 + 366359666251*x^2 - 19410042679742*x + 49980715777967)
 
gp: K = bnfinit(x^14 - 2*x^13 - 29*x^12 + 2684*x^11 - 50815*x^10 - 101186*x^9 + 2905773*x^8 - 96986796*x^7 + 393809153*x^6 + 2822730482*x^5 - 59621810879*x^4 + 536849629536*x^3 + 366359666251*x^2 - 19410042679742*x + 49980715777967, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} - 29 x^{12} + 2684 x^{11} - 50815 x^{10} - 101186 x^{9} + 2905773 x^{8} - 96986796 x^{7} + 393809153 x^{6} + 2822730482 x^{5} - 59621810879 x^{4} + 536849629536 x^{3} + 366359666251 x^{2} - 19410042679742 x + 49980715777967 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1136553186048642579467300999397100786548736=-\,2^{27}\cdot 7^{12}\cdot 409^{5}\cdot 3767^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1009.18$
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:  $2, 7, 409, 3767$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{71352266169663914758674862715139169531800591645594074507187713042850444135} a^{13} - \frac{1427532808034239234278154596262275638203513720050993673988396730642790024}{23784088723221304919558287571713056510600197215198024835729237680950148045} a^{12} - \frac{4288554812861672907272311236528952607839781916394553081490640440005959675}{14270453233932782951734972543027833906360118329118814901437542608570088827} a^{11} + \frac{31276146609805313448244278374197049316841461744947704573000788932142772927}{71352266169663914758674862715139169531800591645594074507187713042850444135} a^{10} - \frac{30679472293474470161272196511630171592770278322680151729299202849515216198}{71352266169663914758674862715139169531800591645594074507187713042850444135} a^{9} + \frac{1106779584288718540168137196369737244480119317720868539500271385794186629}{4756817744644260983911657514342611302120039443039604967145847536190029609} a^{8} - \frac{142494935167691986026519271158060536205278497977838299687156473015213549}{365909057280327767993204424180200869393849187926123459011219041245386893} a^{7} - \frac{3618567004156292242883528195056447324566416511444924546099046886030424436}{23784088723221304919558287571713056510600197215198024835729237680950148045} a^{6} + \frac{6031162702047567366075236757306834848501306263211924794338654159785436127}{14270453233932782951734972543027833906360118329118814901437542608570088827} a^{5} + \frac{257306682699833436764737808861582191355222694838243648428508583371284022}{1829545286401638839966022120901004346969245939630617295056095206226934465} a^{4} - \frac{1720124025939491751162487434093189616961803054604014151296455962175871153}{14270453233932782951734972543027833906360118329118814901437542608570088827} a^{3} - \frac{32496593467852552279699353552934321387866076140108592587758120448399254581}{71352266169663914758674862715139169531800591645594074507187713042850444135} a^{2} - \frac{24264426908038986175170293650701848608219584189889798942556442935797347307}{71352266169663914758674862715139169531800591645594074507187713042850444135} a + \frac{324158878384024000226577117328271249350502187220099975118529521417955074}{5488635859204916519898066362703013040907737818891851885168285618680803395}$

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:  $10$
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:  \( 23006186951600000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T45:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 3528
The 35 conjugacy class representatives for [F_42(7)^2]2=F_42(7)wr2
Character table for [F_42(7)^2]2=F_42(7)wr2 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \)

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

Sibling fields

Degree 28 siblings: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.14.27.213$x^{14} + 2 x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{8} + 4 x^{6} + 4 x^{5} + 4 x^{2} - 2$$14$$1$$27$$(C_7:C_3) \times C_2$$[3]_{7}^{3}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.7.7.3$x^{7} + 35 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
409Data not computed
3767Data not computed