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Properties

Label 14.0.32918715676...4443.2
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{24}\cdot 43^{13}$
Root discriminant $923.70$
Ramified primes $7, 43$
Class number $2756257$ (GRH)
Class group $[2756257]$ (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![2771555675837, 2126473677202, 1042967916088, 366737247170, 94817302349, 18547361140, 2773555771, 307265874, 24550764, 1193766, -24080, -5719, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 5719*x^11 - 24080*x^10 + 1193766*x^9 + 24550764*x^8 + 307265874*x^7 + 2773555771*x^6 + 18547361140*x^5 + 94817302349*x^4 + 366737247170*x^3 + 1042967916088*x^2 + 2126473677202*x + 2771555675837)
 
gp: K = bnfinit(x^14 - 5719*x^11 - 24080*x^10 + 1193766*x^9 + 24550764*x^8 + 307265874*x^7 + 2773555771*x^6 + 18547361140*x^5 + 94817302349*x^4 + 366737247170*x^3 + 1042967916088*x^2 + 2126473677202*x + 2771555675837, 1)
 

Normalized defining polynomial

\( x^{14} - 5719 x^{11} - 24080 x^{10} + 1193766 x^{9} + 24550764 x^{8} + 307265874 x^{7} + 2773555771 x^{6} + 18547361140 x^{5} + 94817302349 x^{4} + 366737247170 x^{3} + 1042967916088 x^{2} + 2126473677202 x + 2771555675837 \)

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:  \(-329187156767051876908796215815371233484443=-\,7^{24}\cdot 43^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $923.70$
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:  $7, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2107=7^{2}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{2107}(512,·)$, $\chi_{2107}(1,·)$, $\chi_{2107}(1282,·)$, $\chi_{2107}(1828,·)$, $\chi_{2107}(1317,·)$, $\chi_{2107}(8,·)$, $\chi_{2107}(64,·)$, $\chi_{2107}(1163,·)$, $\chi_{2107}(428,·)$, $\chi_{2107}(876,·)$, $\chi_{2107}(687,·)$, $\chi_{2107}(1107,·)$, $\chi_{2107}(1982,·)$, $\chi_{2107}(1989,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{13} a^{8} - \frac{2}{13} a^{7} + \frac{4}{13} a^{6} + \frac{5}{13} a^{5} + \frac{4}{13} a^{4} + \frac{5}{13} a^{3} + \frac{3}{13} a^{2} - \frac{6}{13} a$, $\frac{1}{13} a^{9} + \frac{1}{13} a^{5} + \frac{1}{13} a$, $\frac{1}{13} a^{10} + \frac{1}{13} a^{6} + \frac{1}{13} a^{2}$, $\frac{1}{13} a^{11} + \frac{1}{13} a^{7} + \frac{1}{13} a^{3}$, $\frac{1}{1695577} a^{12} - \frac{11186}{1695577} a^{11} + \frac{62453}{1695577} a^{10} - \frac{21752}{1695577} a^{9} - \frac{35290}{1695577} a^{8} - \frac{664899}{1695577} a^{7} + \frac{114027}{1695577} a^{6} - \frac{268030}{1695577} a^{5} - \frac{73134}{1695577} a^{4} - \frac{598402}{1695577} a^{3} - \frac{27905}{130429} a^{2} - \frac{1132}{1695577} a + \frac{41009}{130429}$, $\frac{1}{22780877273031902839983429630931174632031195763845832476748359573} a^{13} - \frac{5649452022120163728742114419566533775127801164179014340626}{22780877273031902839983429630931174632031195763845832476748359573} a^{12} + \frac{790860307361822902637083946220074907510054648594001428931753740}{22780877273031902839983429630931174632031195763845832476748359573} a^{11} - \frac{1097504034322763097006915279559036333609702423235554697710968}{134798090372969839289842778881249554035687548898496050158274317} a^{10} + \frac{172197319205763609122383551123509820919523809350650185774649805}{22780877273031902839983429630931174632031195763845832476748359573} a^{9} - \frac{110645899459853094090467219631448806775574407697206352695241679}{22780877273031902839983429630931174632031195763845832476748359573} a^{8} - \frac{2804230349429271556074158720732744858511881165539777194141016883}{22780877273031902839983429630931174632031195763845832476748359573} a^{7} - \frac{4045908591117165423078413339382680822445443722218006942615910242}{22780877273031902839983429630931174632031195763845832476748359573} a^{6} + \frac{4360110175602852316680947386231374387641442769074628706162142081}{22780877273031902839983429630931174632031195763845832476748359573} a^{5} + \frac{605641152002385647835628102387939257512777892209180934959566503}{1752375174848607910767956125456244202463938135680448652057566121} a^{4} - \frac{9579997883747369639118741020352052861597442003877851940054730647}{22780877273031902839983429630931174632031195763845832476748359573} a^{3} - \frac{5404828541007400351364532856105307203934740329173110017010739330}{22780877273031902839983429630931174632031195763845832476748359573} a^{2} + \frac{3950531920647722852896268012673626210207793051780783553325764111}{22780877273031902839983429630931174632031195763845832476748359573} a + \frac{376927034464835947771744410840682604911643188514219927896808895}{1752375174848607910767956125456244202463938135680448652057566121}$

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

Class group and class number

$C_{2756257}$, which has order $2756257$ (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:  \( 534504501.8334233 \) (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

\(\Q(\sqrt{-43}) \), 7.7.87495801462998035849.4

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{2}$

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
$7$7.14.24.53$x^{14} + 931 x^{13} + 2310 x^{12} + 903 x^{11} + 392 x^{10} + 2198 x^{9} + 2296 x^{8} + 1485 x^{7} + 637 x^{6} + 1295 x^{5} + 2303 x^{4} + 1449 x^{3} + 1316 x^{2} + 2219 x + 2383$$7$$2$$24$$C_{14}$$[2]^{2}$
$43$43.14.13.14$x^{14} + 109300230618147$$14$$1$$13$$C_{14}$$[\ ]_{14}$