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Properties

Label 14.0.57154256495...2439.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 7^{25}\cdot 11^{7}$
Root discriminant $185.51$
Ramified primes $3, 7, 11$
Class number $381036$ (GRH)
Class group $[2, 190518]$ (GRH)
Galois group $C_{14}$ (as 14T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![400484527, -383621049, 374349388, -300869401, 204461971, -93323468, 26899726, -6345898, 1381653, -139706, 34559, -756, 336, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 + 336*x^12 - 756*x^11 + 34559*x^10 - 139706*x^9 + 1381653*x^8 - 6345898*x^7 + 26899726*x^6 - 93323468*x^5 + 204461971*x^4 - 300869401*x^3 + 374349388*x^2 - 383621049*x + 400484527)
 
gp: K = bnfinit(x^14 + 336*x^12 - 756*x^11 + 34559*x^10 - 139706*x^9 + 1381653*x^8 - 6345898*x^7 + 26899726*x^6 - 93323468*x^5 + 204461971*x^4 - 300869401*x^3 + 374349388*x^2 - 383621049*x + 400484527, 1)
 

Normalized defining polynomial

\( x^{14} + 336 x^{12} - 756 x^{11} + 34559 x^{10} - 139706 x^{9} + 1381653 x^{8} - 6345898 x^{7} + 26899726 x^{6} - 93323468 x^{5} + 204461971 x^{4} - 300869401 x^{3} + 374349388 x^{2} - 383621049 x + 400484527 \)

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:  \(-57154256495392789026772590782439=-\,3^{7}\cdot 7^{25}\cdot 11^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $185.51$
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:  $3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1617=3\cdot 7^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1617}(1,·)$, $\chi_{1617}(1154,·)$, $\chi_{1617}(1156,·)$, $\chi_{1617}(230,·)$, $\chi_{1617}(232,·)$, $\chi_{1617}(1385,·)$, $\chi_{1617}(1387,·)$, $\chi_{1617}(461,·)$, $\chi_{1617}(463,·)$, $\chi_{1617}(1616,·)$, $\chi_{1617}(692,·)$, $\chi_{1617}(694,·)$, $\chi_{1617}(923,·)$, $\chi_{1617}(925,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{19} a^{12} - \frac{3}{19} a^{11} - \frac{3}{19} a^{10} - \frac{7}{19} a^{9} - \frac{1}{19} a^{8} + \frac{8}{19} a^{7} - \frac{7}{19} a^{6} - \frac{1}{19} a^{5} + \frac{8}{19} a^{4} - \frac{8}{19} a^{3} + \frac{2}{19} a^{2} - \frac{8}{19} a$, $\frac{1}{63940198442179426129939496143823412219299924941987709883241} a^{13} - \frac{353123196533850355896584240166921364289286742913848637390}{63940198442179426129939496143823412219299924941987709883241} a^{12} - \frac{12406397136930340634158060253131793056408453256899707558618}{63940198442179426129939496143823412219299924941987709883241} a^{11} + \frac{19733004951695341144527643277142183141793663175193041586986}{63940198442179426129939496143823412219299924941987709883241} a^{10} - \frac{8225967706848710507685978527197756371261777063445239067768}{63940198442179426129939496143823412219299924941987709883241} a^{9} + \frac{18826671435609993821937654837836285616389880121508634100162}{63940198442179426129939496143823412219299924941987709883241} a^{8} - \frac{16667057963422405866605848044474941810970313746775050517273}{63940198442179426129939496143823412219299924941987709883241} a^{7} - \frac{13091045086674513141917904391647220124078957170545864009884}{63940198442179426129939496143823412219299924941987709883241} a^{6} - \frac{30744088134126627466730929489553032699305455292755405766755}{63940198442179426129939496143823412219299924941987709883241} a^{5} - \frac{29226284704901805152687107968562480719081387117606791020332}{63940198442179426129939496143823412219299924941987709883241} a^{4} - \frac{9077235348521607763929477507005799180359122726401187334430}{63940198442179426129939496143823412219299924941987709883241} a^{3} - \frac{13771254530016500602569541575625664057820822129582022893040}{63940198442179426129939496143823412219299924941987709883241} a^{2} + \frac{2463015645691906513407065160047850104390088716462351083278}{63940198442179426129939496143823412219299924941987709883241} a - \frac{10066350215923791003470880767962195748287759672159267699}{50227964212238355168844851644794510777140553764326559217}$

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

Class group and class number

$C_{2}\times C_{190518}$, which has order $381036$ (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:  \( 35256.68973693789 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{14}$ (as 14T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{-231}) \), 7.7.13841287201.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R R ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.14.0.1}{14} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$3$3.14.7.1$x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$7$7.14.25.75$x^{14} + 112 x^{13} + 28 x^{12} - 98 x^{11} + 147 x^{10} + 98 x^{9} + 49 x^{8} - 161 x^{7} + 49 x^{6} + 147 x^{5} + 147 x^{2} - 147 x - 126$$14$$1$$25$$C_{14}$$[2]_{2}$
$11$11.14.7.1$x^{14} - 2662 x^{8} + 1771561 x^{2} - 311794736$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$