↑

Properties

Label 14.0.29767262731...9631.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,911^{13}$
Root discriminant $559.92$
Ramified prime $911$
Class number $161479$ (GRH)
Class group $[161479]$ (GRH)
Galois group $C_{14}$ (as 14T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5270284469, -8581811296, 6431287949, -2748137801, 869735265, -265496895, 107210133, -12988271, 1749062, -61103, 10483, -1441, 33, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 + 33*x^12 - 1441*x^11 + 10483*x^10 - 61103*x^9 + 1749062*x^8 - 12988271*x^7 + 107210133*x^6 - 265496895*x^5 + 869735265*x^4 - 2748137801*x^3 + 6431287949*x^2 - 8581811296*x + 5270284469)
 
gp: K = bnfinit(x^14 - x^13 + 33*x^12 - 1441*x^11 + 10483*x^10 - 61103*x^9 + 1749062*x^8 - 12988271*x^7 + 107210133*x^6 - 265496895*x^5 + 869735265*x^4 - 2748137801*x^3 + 6431287949*x^2 - 8581811296*x + 5270284469, 1)
 

Normalized defining polynomial

\( x^{14} - x^{13} + 33 x^{12} - 1441 x^{11} + 10483 x^{10} - 61103 x^{9} + 1749062 x^{8} - 12988271 x^{7} + 107210133 x^{6} - 265496895 x^{5} + 869735265 x^{4} - 2748137801 x^{3} + 6431287949 x^{2} - 8581811296 x + 5270284469 \)

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:  \(-297672627318406929279256927821316759631=-\,911^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $559.92$
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:  $911$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(911\)
Dirichlet character group:    $\lbrace$$\chi_{911}(1,·)$, $\chi_{911}(130,·)$, $\chi_{911}(579,·)$, $\chi_{911}(7,·)$, $\chi_{911}(904,·)$, $\chi_{911}(332,·)$, $\chi_{911}(781,·)$, $\chi_{911}(910,·)$, $\chi_{911}(49,·)$, $\chi_{911}(502,·)$, $\chi_{911}(343,·)$, $\chi_{911}(568,·)$, $\chi_{911}(409,·)$, $\chi_{911}(862,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{437} a^{11} - \frac{156}{437} a^{10} + \frac{191}{437} a^{9} - \frac{132}{437} a^{8} - \frac{55}{437} a^{7} - \frac{110}{437} a^{6} + \frac{160}{437} a^{5} - \frac{212}{437} a^{4} + \frac{13}{437} a^{3} - \frac{68}{437} a^{2} + \frac{162}{437} a - \frac{7}{19}$, $\frac{1}{13547} a^{12} - \frac{7}{13547} a^{11} + \frac{5789}{13547} a^{10} - \frac{3137}{13547} a^{9} - \frac{1806}{13547} a^{8} - \frac{1750}{13547} a^{7} + \frac{6494}{13547} a^{6} - \frac{3903}{13547} a^{5} - \frac{3607}{13547} a^{4} - \frac{3375}{13547} a^{3} + \frac{6199}{13547} a^{2} + \frac{125}{437} a + \frac{154}{589}$, $\frac{1}{117467968421640139188037720148325174400008750212998740704543} a^{13} + \frac{3954363032386756772182359338383481405861354533279710639}{117467968421640139188037720148325174400008750212998740704543} a^{12} - \frac{100589479415741138384854873331502515524368103364491840187}{117467968421640139188037720148325174400008750212998740704543} a^{11} + \frac{26189278928226208489678099945279817415933677086768425148469}{117467968421640139188037720148325174400008750212998740704543} a^{10} - \frac{55868947298828494204158321658870600536844156249711444333025}{117467968421640139188037720148325174400008750212998740704543} a^{9} + \frac{17999611560456475768681324237864308323232500933608468864920}{117467968421640139188037720148325174400008750212998740704543} a^{8} - \frac{18738554361145755415108557447673563564756163634523381631968}{117467968421640139188037720148325174400008750212998740704543} a^{7} + \frac{31805910214055212461731404788187890083503394697934492614385}{117467968421640139188037720148325174400008750212998740704543} a^{6} + \frac{5476529962094856538260626810875252916706081880262487430097}{117467968421640139188037720148325174400008750212998740704543} a^{5} + \frac{16438587633349820114654065616761394188681418044697952742281}{117467968421640139188037720148325174400008750212998740704543} a^{4} - \frac{40918152787278843158436503884322518758937422898582968857337}{117467968421640139188037720148325174400008750212998740704543} a^{3} - \frac{34133574838338865375445889347678197946847500015070783897184}{117467968421640139188037720148325174400008750212998740704543} a^{2} - \frac{10948609945554525070987504427355833340606256758935255307786}{117467968421640139188037720148325174400008750212998740704543} a + \frac{106664689697656874072421905262416760695717810986342918214}{5107302974853919095132074789057616278261250009260814813241}$

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

Class group and class number

$C_{161479}$, which has order $161479$ (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:  \( 504165333.8519378 \) (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

\(\Q(\sqrt{-911}) \), 7.7.571623746239596961.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/29.14.0.1}{14} }$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
911Data not computed