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Properties

Label 15.7.23935035566...0000.1
Degree $15$
Signature $[7, 4]$
Discriminant $2^{14}\cdot 3^{14}\cdot 5^{14}\cdot 7^{10}\cdot 11^{6}$
Root discriminant $228.35$
Ramified primes $2, 3, 5, 7, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T103

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![68779600, 167327200, 92482920, 1819820, 1379896, 4121184, 79390, -44745, 63495, -6401, -2999, 90, 155, -35, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 6*x^14 - 35*x^13 + 155*x^12 + 90*x^11 - 2999*x^10 - 6401*x^9 + 63495*x^8 - 44745*x^7 + 79390*x^6 + 4121184*x^5 + 1379896*x^4 + 1819820*x^3 + 92482920*x^2 + 167327200*x + 68779600)
 
gp: K = bnfinit(x^15 - 6*x^14 - 35*x^13 + 155*x^12 + 90*x^11 - 2999*x^10 - 6401*x^9 + 63495*x^8 - 44745*x^7 + 79390*x^6 + 4121184*x^5 + 1379896*x^4 + 1819820*x^3 + 92482920*x^2 + 167327200*x + 68779600, 1)
 

Normalized defining polynomial

\( x^{15} - 6 x^{14} - 35 x^{13} + 155 x^{12} + 90 x^{11} - 2999 x^{10} - 6401 x^{9} + 63495 x^{8} - 44745 x^{7} + 79390 x^{6} + 4121184 x^{5} + 1379896 x^{4} + 1819820 x^{3} + 92482920 x^{2} + 167327200 x + 68779600 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(239350355667544208264100000000000000=2^{14}\cdot 3^{14}\cdot 5^{14}\cdot 7^{10}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $228.35$
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:  $2, 3, 5, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{10} + \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3} a$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{36} a^{12} + \frac{1}{36} a^{10} + \frac{1}{36} a^{9} - \frac{1}{3} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{5}{12} a^{5} - \frac{5}{12} a^{4} - \frac{2}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{9} a - \frac{2}{9}$, $\frac{1}{60480} a^{13} - \frac{253}{30240} a^{12} - \frac{337}{12096} a^{11} - \frac{761}{12096} a^{10} + \frac{23}{189} a^{9} + \frac{3299}{6720} a^{8} - \frac{4447}{20160} a^{7} - \frac{59}{576} a^{6} + \frac{107}{4032} a^{5} + \frac{733}{1512} a^{4} + \frac{12767}{30240} a^{3} - \frac{509}{3780} a^{2} - \frac{19}{378} a + \frac{685}{1512}$, $\frac{1}{895389364447577710227261842002682290230991680} a^{14} + \frac{604856349929029418752383045991952658019}{179077872889515542045452368400536458046198336} a^{13} + \frac{7437179505472649015104680342244054043920049}{895389364447577710227261842002682290230991680} a^{12} - \frac{718235271944644260446109807544385006515027}{29846312148252590340908728066756076341033056} a^{11} + \frac{627742132592292137638844223964572154932005}{59692624296505180681817456133512152682066112} a^{10} + \frac{7520936454268963911445637165270752105987531}{895389364447577710227261842002682290230991680} a^{9} + \frac{7134790340317669356449594363257441251796037}{29846312148252590340908728066756076341033056} a^{8} + \frac{17521943236343547139038018497603440256966981}{37307890185315737926135910083445095426291320} a^{7} - \frac{4714404846574684865340423897264000306688631}{9948770716084196780302909355585358780344352} a^{6} + \frac{86291150846181931165648064951393547574050209}{179077872889515542045452368400536458046198336} a^{5} - \frac{8653838277279219723160272357439653965437379}{63956383174826979301947274428763020730785120} a^{4} + \frac{23707247837794467505319614672401247134864899}{89538936444757771022726184200268229023099168} a^{3} + \frac{76875918247005247262095628757589395159049}{296094366550124904175681826059088058938820} a^{2} - \frac{613609508325202871123944451673328326161327}{2487192679021049195075727338896339695086088} a - \frac{4639258368554048273697199932592849858337891}{22384734111189442755681546050067057255774792}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $10$
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:  \( 67854593545900 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T103:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 653837184000
The 94 conjugacy class representatives for A15 are not computed
Character table for A15 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 R ${\href{/LocalNumberField/13.11.0.1}{11} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $15$ ${\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ $15$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $15$ $15$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.3.3.1$x^{3} + 6 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.6.11.6$x^{6} + 6 x^{3} + 15$$6$$1$$11$$S_3^2$$[2, 5/2]_{2}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.3.3$x^{4} + 10$$4$$1$$3$$C_4$$[\ ]_{4}$
5.10.11.6$x^{10} + 10 x^{2} + 10$$10$$1$$11$$F_5$$[5/4]_{4}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.3.1$x^{4} + 14$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.8.6.3$x^{8} - 7 x^{4} + 147$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.3.2.1$x^{3} - 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$