↑

Properties

Label 14.0.53588606915...5607.5
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 43^{12}$
Root discriminant $811.37$
Ramified primes $7, 43$
Class number $792673$ (GRH)
Class group $[7, 7, 16177]$ (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![79626326570317, -10557595949299, -2822541592794, 50016502081, 104029582531, 1870004640, -1425878538, -107746304, 13716269, 783202, 301, -7224, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7224*x^11 + 301*x^10 + 783202*x^9 + 13716269*x^8 - 107746304*x^7 - 1425878538*x^6 + 1870004640*x^5 + 104029582531*x^4 + 50016502081*x^3 - 2822541592794*x^2 - 10557595949299*x + 79626326570317)
 
gp: K = bnfinit(x^14 - 7224*x^11 + 301*x^10 + 783202*x^9 + 13716269*x^8 - 107746304*x^7 - 1425878538*x^6 + 1870004640*x^5 + 104029582531*x^4 + 50016502081*x^3 - 2822541592794*x^2 - 10557595949299*x + 79626326570317, 1)
 

Normalized defining polynomial

\( x^{14} - 7224 x^{11} + 301 x^{10} + 783202 x^{9} + 13716269 x^{8} - 107746304 x^{7} - 1425878538 x^{6} + 1870004640 x^{5} + 104029582531 x^{4} + 50016502081 x^{3} - 2822541592794 x^{2} - 10557595949299 x + 79626326570317 \)

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:  \(-53588606915566584613059849086223224055607=-\,7^{25}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $811.37$
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$$\chi_{2107}(1632,·)$, $\chi_{2107}(1,·)$, $\chi_{2107}(1478,·)$, $\chi_{2107}(967,·)$, $\chi_{2107}(680,·)$, $\chi_{2107}(1483,·)$, $\chi_{2107}(1420,·)$, $\chi_{2107}(1294,·)$, $\chi_{2107}(176,·)$, $\chi_{2107}(594,·)$, $\chi_{2107}(1301,·)$, $\chi_{2107}(1847,·)$, $\chi_{2107}(1688,·)$, $\chi_{2107}(188,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{43} a^{7}$, $\frac{1}{43} a^{8}$, $\frac{1}{43} a^{9}$, $\frac{1}{1849} a^{10} + \frac{7}{43} a^{6} - \frac{18}{43} a^{5} + \frac{9}{43} a^{4} + \frac{11}{43} a^{3}$, $\frac{1}{1849} a^{11} - \frac{18}{43} a^{6} + \frac{9}{43} a^{5} + \frac{11}{43} a^{4}$, $\frac{1}{19002173} a^{12} + \frac{45}{441911} a^{11} + \frac{31}{441911} a^{10} - \frac{2232}{441911} a^{9} - \frac{4895}{441911} a^{8} + \frac{3981}{441911} a^{7} - \frac{105900}{441911} a^{6} + \frac{177988}{441911} a^{5} - \frac{59}{239} a^{4} + \frac{3093}{10277} a^{3} - \frac{119}{239} a^{2} - \frac{118}{239} a + \frac{118}{239}$, $\frac{1}{122447462633183472384436203763260563455947228892052622802273} a^{13} + \frac{1998027351896531460304065771767567445056340932611071}{122447462633183472384436203763260563455947228892052622802273} a^{12} + \frac{395051036144341251792703427025042840329836734173349026}{2847615410074034241498516366587454964091796020745409832611} a^{11} + \frac{514246311351385373205677864945924291827273679036342712}{2847615410074034241498516366587454964091796020745409832611} a^{10} + \frac{13335624881466572667348536122945902108296025828175656793}{2847615410074034241498516366587454964091796020745409832611} a^{9} + \frac{11678865475378572999694970319048216334133475610240918543}{2847615410074034241498516366587454964091796020745409832611} a^{8} - \frac{15958071940366991197845563962931022930992737682618574397}{2847615410074034241498516366587454964091796020745409832611} a^{7} + \frac{124734972755732238131027501271785187197567869963735876663}{2847615410074034241498516366587454964091796020745409832611} a^{6} + \frac{1094201749622251528004226938862068990179689811370225197073}{2847615410074034241498516366587454964091796020745409832611} a^{5} + \frac{716245376586858710816014247674790368284188247208490046}{66223614187768238174384101548545464281204558621986275177} a^{4} - \frac{24376314462334228685412853861354262954064892013978711021}{66223614187768238174384101548545464281204558621986275177} a^{3} + \frac{602565805494458841893457134398072561893612999650242253}{1540084050878331120334513989501057308865222293534564539} a^{2} - \frac{330502379419609398252818840444261020550193143340615786}{1540084050878331120334513989501057308865222293534564539} a - \frac{288947101058505823694912338326489329606519641570223322}{1540084050878331120334513989501057308865222293534564539}$

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

Class group and class number

$C_{7}\times C_{7}\times C_{16177}$, which has order $792673$ (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:  \( 224277011.0596314 \) (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

\(\Q(\sqrt{-7}) \), 7.7.87495801462998035849.2

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 ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ R ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$7.14.25.59$x^{14} - 168 x^{13} + 70 x^{12} - 147 x^{11} + 147 x^{10} - 98 x^{9} + 49 x^{8} + 168 x^{7} - 49 x^{4} - 147 x^{3} - 49 x^{2} + 98 x + 126$$14$$1$$25$$C_{14}$$[2]_{2}$
$43$43.7.6.2$x^{7} + 387$$7$$1$$6$$C_7$$[\ ]_{7}$
43.7.6.2$x^{7} + 387$$7$$1$$6$$C_7$$[\ ]_{7}$