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Properties

Label 14.0.13364137855...2512.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,2^{21}\cdot 3^{7}\cdot 7^{7}\cdot 29^{12}$
Root discriminant $232.35$
Ramified primes $2, 3, 7, 29$
Class number $17213728$ (GRH)
Class group $[2, 2, 2, 2, 1075858]$ (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![398209394731, -13804563006, 53691865357, -1691924612, 3215339330, -90555574, 111457382, -2706408, 2428667, -47544, 33450, -466, 271, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2*x^13 + 271*x^12 - 466*x^11 + 33450*x^10 - 47544*x^9 + 2428667*x^8 - 2706408*x^7 + 111457382*x^6 - 90555574*x^5 + 3215339330*x^4 - 1691924612*x^3 + 53691865357*x^2 - 13804563006*x + 398209394731)
 
gp: K = bnfinit(x^14 - 2*x^13 + 271*x^12 - 466*x^11 + 33450*x^10 - 47544*x^9 + 2428667*x^8 - 2706408*x^7 + 111457382*x^6 - 90555574*x^5 + 3215339330*x^4 - 1691924612*x^3 + 53691865357*x^2 - 13804563006*x + 398209394731, 1)
 

Normalized defining polynomial

\( x^{14} - 2 x^{13} + 271 x^{12} - 466 x^{11} + 33450 x^{10} - 47544 x^{9} + 2428667 x^{8} - 2706408 x^{7} + 111457382 x^{6} - 90555574 x^{5} + 3215339330 x^{4} - 1691924612 x^{3} + 53691865357 x^{2} - 13804563006 x + 398209394731 \)

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:  \(-1336413785513566222733246057152512=-\,2^{21}\cdot 3^{7}\cdot 7^{7}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $232.35$
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:  $2, 3, 7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4872=2^{3}\cdot 3\cdot 7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{4872}(1,·)$, $\chi_{4872}(3529,·)$, $\chi_{4872}(1763,·)$, $\chi_{4872}(2017,·)$, $\chi_{4872}(4201,·)$, $\chi_{4872}(587,·)$, $\chi_{4872}(923,·)$, $\chi_{4872}(1009,·)$, $\chi_{4872}(83,·)$, $\chi_{4872}(755,·)$, $\chi_{4872}(2771,·)$, $\chi_{4872}(4705,·)$, $\chi_{4872}(169,·)$, $\chi_{4872}(4283,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{17} a^{10} - \frac{7}{17} a^{9} - \frac{2}{17} a^{8} - \frac{2}{17} a^{6} - \frac{2}{17} a^{5} - \frac{7}{17} a^{4} - \frac{3}{17} a^{3} + \frac{5}{17} a^{2} + \frac{7}{17} a$, $\frac{1}{17} a^{11} + \frac{3}{17} a^{8} - \frac{2}{17} a^{7} + \frac{1}{17} a^{6} - \frac{4}{17} a^{5} - \frac{1}{17} a^{4} + \frac{1}{17} a^{3} + \frac{8}{17} a^{2} - \frac{2}{17} a$, $\frac{1}{11849} a^{12} + \frac{4}{11849} a^{11} - \frac{215}{11849} a^{10} - \frac{3796}{11849} a^{9} + \frac{1834}{11849} a^{8} + \frac{3580}{11849} a^{7} - \frac{1865}{11849} a^{6} - \frac{5231}{11849} a^{5} + \frac{30}{289} a^{4} + \frac{89}{289} a^{3} + \frac{5432}{11849} a^{2} + \frac{135}{697} a - \frac{6}{41}$, $\frac{1}{12170445126899968252060737197804181552523049} a^{13} - \frac{194664739351522052621343333372238381458}{12170445126899968252060737197804181552523049} a^{12} - \frac{105425503044806608198662055033460265476724}{12170445126899968252060737197804181552523049} a^{11} + \frac{143663688113983346116251406655791854887119}{12170445126899968252060737197804181552523049} a^{10} + \frac{138842654245874467526663889978168760701333}{12170445126899968252060737197804181552523049} a^{9} + \frac{5828775894537285630159962700004712718256117}{12170445126899968252060737197804181552523049} a^{8} + \frac{2317490984934575121820291813546821148976968}{12170445126899968252060737197804181552523049} a^{7} + \frac{4141073820048540260595811539606038868586934}{12170445126899968252060737197804181552523049} a^{6} + \frac{2384729216661116119135178934164376967280684}{12170445126899968252060737197804181552523049} a^{5} - \frac{74280016891143218362919866779685315578355}{296840125046340689074652126775711745183489} a^{4} - \frac{1836228499124915926451590642003168392655}{715908536876468720709455129282598914854297} a^{3} - \frac{2344785645899964469091551643349030142776985}{12170445126899968252060737197804181552523049} a^{2} + \frac{255926982902904907290198515722670309791844}{715908536876468720709455129282598914854297} a + \frac{53375187641060679543690580483213188523}{220483072644431389192933516871758212151}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{1075858}$, which has order $17213728$ (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:  \( 6020.985100147561 \) (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

\(\Q(\sqrt{-42}) \), 7.7.594823321.1

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 R R ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{14}$

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
$2$2.14.21.40$x^{14} + 8 x^{13} - 4 x^{12} + 4 x^{11} + 5 x^{10} - 4 x^{9} - 4 x^{8} + 2 x^{7} - x^{6} + 6 x^{5} - 4 x^{4} + 6 x^{3} + 3 x^{2} + 6 x + 3$$2$$7$$21$$C_{14}$$[3]^{7}$
$3$3.14.7.1$x^{14} - 54 x^{8} - 243 x^{4} - 729 x^{2} - 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$7$7.14.7.2$x^{14} - 686 x^{8} + 117649 x^{2} - 3294172$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} - 29$$7$$1$$6$$C_7$$[\ ]_{7}$