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Properties

Label 14.0.47448990293...6087.6
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 29^{12}$
Root discriminant $578.88$
Ramified primes $7, 29$
Class number $250047$ (GRH)
Class group $[3, 3, 3, 21, 21, 21]$ (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![7775628764171, 2923659885741, 941525394390, 200176985575, 39308800027, 5616786700, 733767454, 70091666, 5865685, 239134, 8729, -812, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 812*x^11 + 8729*x^10 + 239134*x^9 + 5865685*x^8 + 70091666*x^7 + 733767454*x^6 + 5616786700*x^5 + 39308800027*x^4 + 200176985575*x^3 + 941525394390*x^2 + 2923659885741*x + 7775628764171)
 
gp: K = bnfinit(x^14 - 812*x^11 + 8729*x^10 + 239134*x^9 + 5865685*x^8 + 70091666*x^7 + 733767454*x^6 + 5616786700*x^5 + 39308800027*x^4 + 200176985575*x^3 + 941525394390*x^2 + 2923659885741*x + 7775628764171, 1)
 

Normalized defining polynomial

\( x^{14} - 812 x^{11} + 8729 x^{10} + 239134 x^{9} + 5865685 x^{8} + 70091666 x^{7} + 733767454 x^{6} + 5616786700 x^{5} + 39308800027 x^{4} + 200176985575 x^{3} + 941525394390 x^{2} + 2923659885741 x + 7775628764171 \)

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:  \(-474489902930063357496193840307474416087=-\,7^{25}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $578.88$
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:  $7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1421=7^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1421}(1184,·)$, $\chi_{1421}(1,·)$, $\chi_{1421}(923,·)$, $\chi_{1421}(36,·)$, $\chi_{1421}(993,·)$, $\chi_{1421}(1205,·)$, $\chi_{1421}(1415,·)$, $\chi_{1421}(750,·)$, $\chi_{1421}(1296,·)$, $\chi_{1421}(146,·)$, $\chi_{1421}(83,·)$, $\chi_{1421}(545,·)$, $\chi_{1421}(1147,·)$, $\chi_{1421}(223,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{29} a^{7}$, $\frac{1}{29} a^{8}$, $\frac{1}{29} a^{9}$, $\frac{1}{841} a^{10} + \frac{11}{29} a^{6} + \frac{10}{29} a^{5} - \frac{10}{29} a^{4} + \frac{7}{29} a^{3}$, $\frac{1}{841} a^{11} + \frac{10}{29} a^{6} - \frac{10}{29} a^{5} + \frac{7}{29} a^{4}$, $\frac{1}{24389} a^{12} + \frac{1}{841} a^{9} + \frac{11}{841} a^{8} + \frac{10}{841} a^{7} - \frac{416}{841} a^{6} - \frac{80}{841} a^{5}$, $\frac{1}{2541280357073809278804793349567202819118689370323317740661} a^{13} - \frac{39972097276220982572720100186281608322853653280854174}{2541280357073809278804793349567202819118689370323317740661} a^{12} + \frac{41585925897918151330810013170425396349139326963571246}{87630357140476182027751494812662166176506530011148887609} a^{11} + \frac{10737447681182926676494189881135846362125252782264845}{87630357140476182027751494812662166176506530011148887609} a^{10} - \frac{989995062871455576161717966699935207992573334189059722}{87630357140476182027751494812662166176506530011148887609} a^{9} - \frac{725770893086569442124250285684406857817975844132005712}{87630357140476182027751494812662166176506530011148887609} a^{8} + \frac{1497391535978987409701007201512700513904966431287307141}{87630357140476182027751494812662166176506530011148887609} a^{7} - \frac{33342134865008805009942083246538736929721123804904966383}{87630357140476182027751494812662166176506530011148887609} a^{6} - \frac{28983285220459844055250238529789065790597240220576851281}{87630357140476182027751494812662166176506530011148887609} a^{5} + \frac{511556837798046138251372803197057132618618833908761700}{3021736453119868345784534303884902281948501034867203021} a^{4} - \frac{313915992063962235198168771615699639112236533090506489}{3021736453119868345784534303884902281948501034867203021} a^{3} + \frac{41399233492433513740950057856730517096718398939030445}{104197808728271322268432217375341457998224173616110449} a^{2} - \frac{44880769580970559566386714667709543804112259828117540}{104197808728271322268432217375341457998224173616110449} a + \frac{17269725434396934825496419270820983464603374942721163}{104197808728271322268432217375341457998224173616110449}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{21}\times C_{21}\times C_{21}$, which has order $250047$ (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:  \( 59811086.44175506 \) (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

\(\Q(\sqrt{-7}) \), 7.7.8233120419813614521.6

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$7$7.14.25.52$x^{14} - 56 x^{13} - 112 x^{12} + 49 x^{11} + 98 x^{10} - 147 x^{9} - 42 x^{7} + 98 x^{6} - 49 x^{5} + 98 x^{4} - 98 x^{3} + 147 x - 84$$14$$1$$25$$C_{14}$$[2]_{2}$
$29$29.7.6.3$x^{7} + 928$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.3$x^{7} + 928$$7$$1$$6$$C_7$$[\ ]_{7}$