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Properties

Label 14.0.17192704271...2251.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,3^{7}\cdot 7^{24}\cdot 17^{7}$
Root discriminant $200.69$
Ramified primes $3, 7, 17$
Class number $513914$ (GRH)
Class group $[513914]$ (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![704858827, -296315320, 206001411, -61248166, 25808202, -5829124, 1803081, -312301, 80535, -10829, 2506, -287, 70, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 + 70*x^12 - 287*x^11 + 2506*x^10 - 10829*x^9 + 80535*x^8 - 312301*x^7 + 1803081*x^6 - 5829124*x^5 + 25808202*x^4 - 61248166*x^3 + 206001411*x^2 - 296315320*x + 704858827)
 
gp: K = bnfinit(x^14 - 7*x^13 + 70*x^12 - 287*x^11 + 2506*x^10 - 10829*x^9 + 80535*x^8 - 312301*x^7 + 1803081*x^6 - 5829124*x^5 + 25808202*x^4 - 61248166*x^3 + 206001411*x^2 - 296315320*x + 704858827, 1)
 

Normalized defining polynomial

\( x^{14} - 7 x^{13} + 70 x^{12} - 287 x^{11} + 2506 x^{10} - 10829 x^{9} + 80535 x^{8} - 312301 x^{7} + 1803081 x^{6} - 5829124 x^{5} + 25808202 x^{4} - 61248166 x^{3} + 206001411 x^{2} - 296315320 x + 704858827 \)

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:  \(-171927042716763378495925978132251=-\,3^{7}\cdot 7^{24}\cdot 17^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $200.69$
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:  $3, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2499=3\cdot 7^{2}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{2499}(1,·)$, $\chi_{2499}(715,·)$, $\chi_{2499}(1478,·)$, $\chi_{2499}(358,·)$, $\chi_{2499}(1121,·)$, $\chi_{2499}(1835,·)$, $\chi_{2499}(2192,·)$, $\chi_{2499}(50,·)$, $\chi_{2499}(1429,·)$, $\chi_{2499}(407,·)$, $\chi_{2499}(1072,·)$, $\chi_{2499}(764,·)$, $\chi_{2499}(1786,·)$, $\chi_{2499}(2143,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19} a^{10} - \frac{6}{19} a^{9} + \frac{6}{19} a^{8} + \frac{3}{19} a^{7} - \frac{2}{19} a^{6} - \frac{2}{19} a^{5} + \frac{5}{19} a^{4} + \frac{1}{19} a^{3} - \frac{7}{19} a^{2} - \frac{3}{19} a$, $\frac{1}{19} a^{11} + \frac{8}{19} a^{9} + \frac{1}{19} a^{8} - \frac{3}{19} a^{7} + \frac{5}{19} a^{6} - \frac{7}{19} a^{5} - \frac{7}{19} a^{4} - \frac{1}{19} a^{3} - \frac{7}{19} a^{2} + \frac{1}{19} a$, $\frac{1}{589} a^{12} + \frac{12}{589} a^{11} + \frac{9}{589} a^{10} + \frac{167}{589} a^{9} - \frac{270}{589} a^{8} + \frac{10}{589} a^{7} - \frac{120}{589} a^{6} - \frac{3}{19} a^{5} + \frac{205}{589} a^{4} + \frac{229}{589} a^{3} - \frac{128}{589} a^{2} - \frac{143}{589} a + \frac{4}{31}$, $\frac{1}{116788442723601437805333619931555213359} a^{13} - \frac{33214378038735399352898473458458743}{116788442723601437805333619931555213359} a^{12} - \frac{1062800006457773774502595137428494668}{116788442723601437805333619931555213359} a^{11} + \frac{746713395518820730383253522178416843}{116788442723601437805333619931555213359} a^{10} + \frac{39207527425599865959635214563072652085}{116788442723601437805333619931555213359} a^{9} + \frac{333793897385007022583575956998839220}{116788442723601437805333619931555213359} a^{8} + \frac{757396884309326587391156158698564120}{1743111085426887131422889849724704677} a^{7} - \frac{7677160089413623789172700057108592272}{116788442723601437805333619931555213359} a^{6} + \frac{52865377880265461094482305581782195682}{116788442723601437805333619931555213359} a^{5} + \frac{55568826942608631613211807277042507981}{116788442723601437805333619931555213359} a^{4} + \frac{26000372051739994761082728952888063975}{116788442723601437805333619931555213359} a^{3} - \frac{44815880717578802386631539503801890974}{116788442723601437805333619931555213359} a^{2} - \frac{46128248283477226618152294870618745977}{116788442723601437805333619931555213359} a - \frac{35406546574048041887850191009293370}{91742688706678270074888939459194983}$

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

Class group and class number

$C_{513914}$, which has order $513914$ (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

\(\Q(\sqrt{-51}) \), 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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ R ${\href{/LocalNumberField/5.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/19.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.14.0.1}{14} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$3$3.14.7.1$x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$7$7.14.24.53$x^{14} + 931 x^{13} + 2310 x^{12} + 903 x^{11} + 392 x^{10} + 2198 x^{9} + 2296 x^{8} + 1485 x^{7} + 637 x^{6} + 1295 x^{5} + 2303 x^{4} + 1449 x^{3} + 1316 x^{2} + 2219 x + 2383$$7$$2$$24$$C_{14}$$[2]^{2}$
$17$17.14.7.2$x^{14} - 24137569 x^{2} + 1231016019$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$