↑

Properties

Label 14.0.47448990293...6087.1
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 29^{12}$
Root discriminant $578.88$
Ramified primes $7, 29$
Class number $354368$ (GRH)
Class group $[4, 28, 3164]$ (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![542414506181, 463999676273, 40865281618, -29451595453, -434148589, -1577480520, 215110980, -2313968, 4942035, -45066, 8729, -812, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 812*x^11 + 8729*x^10 - 45066*x^9 + 4942035*x^8 - 2313968*x^7 + 215110980*x^6 - 1577480520*x^5 - 434148589*x^4 - 29451595453*x^3 + 40865281618*x^2 + 463999676273*x + 542414506181)
 
gp: K = bnfinit(x^14 - 812*x^11 + 8729*x^10 - 45066*x^9 + 4942035*x^8 - 2313968*x^7 + 215110980*x^6 - 1577480520*x^5 - 434148589*x^4 - 29451595453*x^3 + 40865281618*x^2 + 463999676273*x + 542414506181, 1)
 

Normalized defining polynomial

\( x^{14} - 812 x^{11} + 8729 x^{10} - 45066 x^{9} + 4942035 x^{8} - 2313968 x^{7} + 215110980 x^{6} - 1577480520 x^{5} - 434148589 x^{4} - 29451595453 x^{3} + 40865281618 x^{2} + 463999676273 x + 542414506181 \)

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$$\chi_{1421}(1,·)$, $\chi_{1421}(741,·)$, $\chi_{1421}(1254,·)$, $\chi_{1421}(1196,·)$, $\chi_{1421}(1357,·)$, $\chi_{1421}(111,·)$, $\chi_{1421}(146,·)$, $\chi_{1421}(1301,·)$, $\chi_{1421}(575,·)$, $\chi_{1421}(953,·)$, $\chi_{1421}(890,·)$, $\chi_{1421}(603,·)$, $\chi_{1421}(190,·)$, $\chi_{1421}(629,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{29} a^{7}$, $\frac{1}{29} a^{8}$, $\frac{1}{29} a^{9}$, $\frac{1}{841} a^{10} + \frac{11}{29} a^{6} + \frac{12}{29} a^{5} + \frac{11}{29} a^{4} - \frac{13}{29} a^{3}$, $\frac{1}{841} a^{11} + \frac{12}{29} a^{6} + \frac{11}{29} a^{5} - \frac{13}{29} a^{4}$, $\frac{1}{101970409} a^{12} + \frac{388}{3516221} a^{11} - \frac{408}{3516221} a^{10} - \frac{9685}{3516221} a^{9} + \frac{33941}{3516221} a^{8} - \frac{26146}{3516221} a^{7} - \frac{463032}{3516221} a^{6} + \frac{1203023}{3516221} a^{5} - \frac{22495}{121249} a^{4} + \frac{28272}{121249} a^{3} + \frac{1512}{4181} a^{2} - \frac{1504}{4181} a + \frac{1996}{4181}$, $\frac{1}{175094130901747822961801700905695134480827425710739199729} a^{13} - \frac{368040854223053593632362534114104453174780548322}{175094130901747822961801700905695134480827425710739199729} a^{12} - \frac{470380663751706496791273258897133233018370895210848}{6037728651784407688337989686403280499338876748646179301} a^{11} - \frac{1173960572355224478344588666024255489390649234044448}{6037728651784407688337989686403280499338876748646179301} a^{10} - \frac{62282235534852023831348173965626469282604097768429211}{6037728651784407688337989686403280499338876748646179301} a^{9} - \frac{3062318350449004855111024262486208937956442448903983}{6037728651784407688337989686403280499338876748646179301} a^{8} - \frac{590387457740565597002007987452354783327449157769277}{163181855453632640225351072605494067549699371585031873} a^{7} + \frac{2654312844081357390422164401705259346924922444028024942}{6037728651784407688337989686403280499338876748646179301} a^{6} - \frac{1575175206696497181524067543540636441723813132410549297}{6037728651784407688337989686403280499338876748646179301} a^{5} + \frac{19783397366848080365404526323880069463403671653252404}{208197539716703713390965161600113120666857818918833769} a^{4} - \frac{47611228425753547840616483259448667433420900659768873}{208197539716703713390965161600113120666857818918833769} a^{3} + \frac{3211901825315238052684736670963488735727008897889593}{7179225507472541841067764193107348988512338583408061} a^{2} + \frac{2351119464258970889226511413934331318085875342038730}{7179225507472541841067764193107348988512338583408061} a - \frac{686294203108597194079428409742358805216276239545412}{7179225507472541841067764193107348988512338583408061}$

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

Class group and class number

$C_{4}\times C_{28}\times C_{3164}$, which has order $354368$ (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:  \( 93363968.92853843 \) (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

\(\Q(\sqrt{-7}) \), 7.7.8233120419813614521.5

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 ${\href{/LocalNumberField/11.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{14}$ ${\href{/LocalNumberField/41.14.0.1}{14} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.14.0.1}{14} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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$7.14.25.95$x^{14} + 168 x^{13} + 140 x^{12} + 49 x^{11} + 49 x^{10} - 49 x^{9} - 168 x^{7} - 147 x^{6} + 147 x^{5} - 49 x^{4} - 98 x^{3} + 98 x^{2} - 98 x - 91$$14$$1$$25$$C_{14}$$[2]_{2}$
$29$29.7.6.6$x^{7} - 464$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.6$x^{7} - 464$$7$$1$$6$$C_7$$[\ ]_{7}$