↑

Properties

Label 14.0.47448990293...6087.4
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 29^{12}$
Root discriminant $578.88$
Ramified primes $7, 29$
Class number $82369$ (GRH)
Class group $[287, 287]$ (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![2558606699525, -433333386675, 50915744628, -28019912375, 10397884315, -2313638096, 382419520, -37003420, 5175079, -221270, 17255, -812, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 812*x^11 + 17255*x^10 - 221270*x^9 + 5175079*x^8 - 37003420*x^7 + 382419520*x^6 - 2313638096*x^5 + 10397884315*x^4 - 28019912375*x^3 + 50915744628*x^2 - 433333386675*x + 2558606699525)
 
gp: K = bnfinit(x^14 - 812*x^11 + 17255*x^10 - 221270*x^9 + 5175079*x^8 - 37003420*x^7 + 382419520*x^6 - 2313638096*x^5 + 10397884315*x^4 - 28019912375*x^3 + 50915744628*x^2 - 433333386675*x + 2558606699525, 1)
 

Normalized defining polynomial

\( x^{14} - 812 x^{11} + 17255 x^{10} - 221270 x^{9} + 5175079 x^{8} - 37003420 x^{7} + 382419520 x^{6} - 2313638096 x^{5} + 10397884315 x^{4} - 28019912375 x^{3} + 50915744628 x^{2} - 433333386675 x + 2558606699525 \)

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:  \(-474489902930063357496193840307474416087=-\,7^{25}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $578.88$
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:  $7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1421=7^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1421}(1,·)$, $\chi_{1421}(1035,·)$, $\chi_{1421}(1156,·)$, $\chi_{1421}(1098,·)$, $\chi_{1421}(1387,·)$, $\chi_{1421}(748,·)$, $\chi_{1421}(335,·)$, $\chi_{1421}(720,·)$, $\chi_{1421}(146,·)$, $\chi_{1421}(596,·)$, $\chi_{1421}(1399,·)$, $\chi_{1421}(484,·)$, $\chi_{1421}(1051,·)$, $\chi_{1421}(1212,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{29} a^{7}$, $\frac{1}{29} a^{8}$, $\frac{1}{29} a^{9}$, $\frac{1}{4205} a^{10} + \frac{1}{145} a^{7} + \frac{3}{29} a^{6} + \frac{11}{29} a^{5} + \frac{72}{145} a^{4} + \frac{4}{29} a^{3} + \frac{2}{5} a$, $\frac{1}{4205} a^{11} + \frac{1}{145} a^{8} + \frac{11}{29} a^{6} + \frac{72}{145} a^{5} + \frac{4}{29} a^{4} + \frac{2}{5} a^{2}$, $\frac{1}{5243635} a^{12} + \frac{1}{180815} a^{11} + \frac{16}{180815} a^{10} + \frac{1306}{180815} a^{9} + \frac{2509}{180815} a^{8} + \frac{2984}{180815} a^{7} + \frac{78517}{180815} a^{6} - \frac{87792}{180815} a^{5} + \frac{882}{6235} a^{4} + \frac{2002}{6235} a^{3} + \frac{72}{215} a^{2} + \frac{12}{215} a - \frac{1}{43}$, $\frac{1}{2164909326228085693876880763200354365285203555925474381555} a^{13} + \frac{1254606188924452151293789745678280064123349353722}{2164909326228085693876880763200354365285203555925474381555} a^{12} - \frac{13082015081710896542719549825884782367412335646327}{514841694703468654905322416932307815763425340291432671} a^{11} + \frac{586086234045345412196038395046739220532615855047691}{14930409146400590992254350091036926657139334868451547459} a^{10} + \frac{1180892679174328394912868977334573447056564435604872532}{74652045732002954961271750455184633285696674342257737295} a^{9} - \frac{60936778147210812815230777676161887680539810373171522}{14930409146400590992254350091036926657139334868451547459} a^{8} - \frac{1274810285464295722200885817553450717554617600556316467}{74652045732002954961271750455184633285696674342257737295} a^{7} + \frac{18351505428809861086369238324743523541203635215281315654}{74652045732002954961271750455184633285696674342257737295} a^{6} - \frac{3504327978772254080842761351949895355712567259596787276}{14930409146400590992254350091036926657139334868451547459} a^{5} + \frac{5517409969038340116245219671017022596727502713443381}{514841694703468654905322416932307815763425340291432671} a^{4} + \frac{37044729456874697456424477646206049603840344730168869}{2574208473517343274526612084661539078817126701457163355} a^{3} - \frac{1021741776685650153607473155448697627195834700357345}{17753161886326505341562841963183028129773287596256299} a^{2} + \frac{27103125437615895741778931681677136014235603046239403}{88765809431632526707814209815915140648866437981281495} a - \frac{4290562508356437616046091329919604561338427457108586}{17753161886326505341562841963183028129773287596256299}$

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

Class group and class number

$C_{287}\times C_{287}$, which has order $82369$ (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:  \( 368248548.12946504 \) (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

\(\Q(\sqrt{-7}) \), 7.7.8233120419813614521.2

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/3.14.0.1}{14} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$7.14.25.72$x^{14} + 63 x^{13} + 35 x^{12} + 98 x^{11} + 98 x^{10} - 98 x^{9} + 147 x^{8} - 112 x^{7} - 49 x^{6} + 147 x^{5} + 98 x^{4} - 49 x^{3} + 147 x^{2} + 49 x + 63$$14$$1$$25$$C_{14}$$[2]_{2}$
$29$29.7.6.1$x^{7} + 232$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.1$x^{7} + 232$$7$$1$$6$$C_7$$[\ ]_{7}$