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Properties

Label 14.14.1208246280...0000.1
Degree $14$
Signature $[14, 0]$
Discriminant $2^{14}\cdot 3^{22}\cdot 5^{12}\cdot 7^{14}\cdot 19^{6}\cdot 197^{6}\cdot 199^{6}\cdot 2593^{6}\cdot 34963^{6}\cdot 560233^{6}\cdot 1072255949^{2}\cdot 6488895781911832501^{2}$
Root discriminant $729{,}476{,}162{,}501{,}298.53$
Ramified primes $2, 3, 5, 7, 19, 197, 199, 2593, 34963, 560233, 1072255949, 6488895781911832501$
Class number Not computed
Class group Not computed
Galois group 14T62

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![40095511130962225240189804491943359375, 773043792817583070706838466796875000, 2587399700316109318860370458984375, -18637303238230533691855781250000, -63938673030863212150078125000, 144716607884926457572500000, 517062546566176619531250, -447630993322011000000, -1855964346085631250, 597401557200000, 3313325898225, -287910000, -2900730, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2900730*x^12 - 287910000*x^11 + 3313325898225*x^10 + 597401557200000*x^9 - 1855964346085631250*x^8 - 447630993322011000000*x^7 + 517062546566176619531250*x^6 + 144716607884926457572500000*x^5 - 63938673030863212150078125000*x^4 - 18637303238230533691855781250000*x^3 + 2587399700316109318860370458984375*x^2 + 773043792817583070706838466796875000*x + 40095511130962225240189804491943359375)
 
gp: K = bnfinit(x^14 - 2900730*x^12 - 287910000*x^11 + 3313325898225*x^10 + 597401557200000*x^9 - 1855964346085631250*x^8 - 447630993322011000000*x^7 + 517062546566176619531250*x^6 + 144716607884926457572500000*x^5 - 63938673030863212150078125000*x^4 - 18637303238230533691855781250000*x^3 + 2587399700316109318860370458984375*x^2 + 773043792817583070706838466796875000*x + 40095511130962225240189804491943359375, 1)
 

Normalized defining polynomial

\( x^{14} - 2900730 x^{12} - 287910000 x^{11} + 3313325898225 x^{10} + 597401557200000 x^{9} - 1855964346085631250 x^{8} - 447630993322011000000 x^{7} + 517062546566176619531250 x^{6} + 144716607884926457572500000 x^{5} - 63938673030863212150078125000 x^{4} - 18637303238230533691855781250000 x^{3} + 2587399700316109318860370458984375 x^{2} + 773043792817583070706838466796875000 x + 40095511130962225240189804491943359375 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12082462805952683578833887544607885354840529275474033703346853825890523525763932307253207072895617326779140942034887593472026784080351332996854475050975704479877705399512513606726026903633577245444000000000000=2^{14}\cdot 3^{22}\cdot 5^{12}\cdot 7^{14}\cdot 19^{6}\cdot 197^{6}\cdot 199^{6}\cdot 2593^{6}\cdot 34963^{6}\cdot 560233^{6}\cdot 1072255949^{2}\cdot 6488895781911832501^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $729{,}476{,}162{,}501{,}298.53$
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:  $2, 3, 5, 7, 19, 197, 199, 2593, 34963, 560233, 1072255949, 6488895781911832501$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{25} a^{3} + \frac{2}{5} a$, $\frac{1}{125} a^{4} + \frac{2}{25} a^{2}$, $\frac{1}{625} a^{5} + \frac{2}{125} a^{3}$, $\frac{1}{3125} a^{6} + \frac{2}{625} a^{4}$, $\frac{1}{244140625} a^{7} + \frac{22427}{48828125} a^{5} + \frac{922}{390625} a^{4} - \frac{4938}{390625} a^{3} - \frac{3542}{78125} a^{2} - \frac{877}{3125} a + \frac{2982}{15625}$, $\frac{1}{57373046875} a^{8} + \frac{23}{11474609375} a^{7} + \frac{569302}{11474609375} a^{6} + \frac{851371}{2294921875} a^{5} + \frac{144842}{91796875} a^{4} - \frac{201491}{18359375} a^{3} + \frac{242274}{3671875} a^{2} + \frac{1498707}{3671875} a + \frac{84211}{734375}$, $\frac{1}{286865234375} a^{9} + \frac{72}{57373046875} a^{7} - \frac{49078}{458984375} a^{6} - \frac{920807}{2294921875} a^{5} + \frac{35148}{91796875} a^{4} + \frac{289038}{18359375} a^{3} + \frac{1340642}{18359375} a^{2} + \frac{67317}{146875} a + \frac{167478}{734375}$, $\frac{1}{1434326171875} a^{10} + \frac{2}{286865234375} a^{8} + \frac{4}{11474609375} a^{7} - \frac{309457}{2294921875} a^{6} - \frac{1539472}{2294921875} a^{5} - \frac{14473}{18359375} a^{4} - \frac{767403}{91796875} a^{3} + \frac{231177}{3671875} a^{2} + \frac{5398}{15625} a + \frac{108658}{734375}$, $\frac{1}{7171630859375} a^{11} + \frac{2}{1434326171875} a^{9} - \frac{7}{11474609375} a^{7} - \frac{28961}{2294921875} a^{6} - \frac{12101}{18359375} a^{5} - \frac{1705518}{458984375} a^{4} + \frac{3749}{3671875} a^{3} - \frac{103779}{3671875} a^{2} + \frac{221721}{734375} a + \frac{13696}{29375}$, $\frac{1}{107574462890625} a^{12} - \frac{1}{7171630859375} a^{10} + \frac{1}{286865234375} a^{8} + \frac{2}{2294921875} a^{7} + \frac{1529692}{11474609375} a^{6} - \frac{268177}{2294921875} a^{5} - \frac{352148}{91796875} a^{4} - \frac{67199}{91796875} a^{3} + \frac{16308}{3671875} a^{2} + \frac{1200367}{3671875} a - \frac{303731}{734375}$, $\frac{1}{107574462890625} a^{13} + \frac{2}{1434326171875} a^{9} - \frac{67}{57373046875} a^{7} + \frac{130232}{2294921875} a^{6} - \frac{364836}{2294921875} a^{5} + \frac{1629172}{458984375} a^{4} + \frac{333564}{18359375} a^{3} + \frac{660938}{18359375} a^{2} - \frac{11863}{29375} a - \frac{186516}{734375}$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

14T62:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 43589145600
The 72 conjugacy class representatives for A14 are not computed
Character table for A14 is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

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 R R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ R ${\href{/LocalNumberField/23.11.0.1}{11} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.13.0.1}{13} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.13.0.1}{13} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.12.12.21$x^{12} + 44 x^{10} + 45 x^{8} - 48 x^{6} + 59 x^{4} - 60 x^{2} + 23$$2$$6$$12$12T134$[2, 2, 2, 2, 2, 2]^{6}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.12.21.2$x^{12} - 3 x^{11} - 6 x^{10} + 9 x^{7} + 12 x^{6} - 9 x^{5} + 9 x^{3} - 9 x^{2} + 9 x + 6$$12$$1$$21$12T169$[3/2, 2, 9/4, 9/4]_{4}^{2}$
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.5.5.4$x^{5} + 10 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.5.5.4$x^{5} + 10 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
$7$7.7.7.6$x^{7} + 28 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
7.7.7.5$x^{7} + 7 x + 7$$7$$1$$7$$F_7$$[7/6]_{6}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.12.6.1$x^{12} + 41154 x^{6} - 2476099 x^{2} + 423412929$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
197Data not computed
$199$$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
199.12.6.1$x^{12} + 236417970 x^{6} - 312079600999 x^{2} + 13973364134730225$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
2593Data not computed
34963Data not computed
560233Data not computed
1072255949Data not computed
6488895781911832501Data not computed