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Properties

Label 15.3.16524388251...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{30}\cdot 3^{20}\cdot 5^{6}\cdot 7^{10}$
Root discriminant $120.56$
Ramified primes $2, 3, 5, 7$
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![-265728, -218688, -727680, -505624, 360336, 330672, -3072, -42786, -7704, 3052, 864, -342, -232, -42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 42*x^13 - 232*x^12 - 342*x^11 + 864*x^10 + 3052*x^9 - 7704*x^8 - 42786*x^7 - 3072*x^6 + 330672*x^5 + 360336*x^4 - 505624*x^3 - 727680*x^2 - 218688*x - 265728)
 
gp: K = bnfinit(x^15 - 42*x^13 - 232*x^12 - 342*x^11 + 864*x^10 + 3052*x^9 - 7704*x^8 - 42786*x^7 - 3072*x^6 + 330672*x^5 + 360336*x^4 - 505624*x^3 - 727680*x^2 - 218688*x - 265728, 1)
 

Normalized defining polynomial

\( x^{15} - 42 x^{13} - 232 x^{12} - 342 x^{11} + 864 x^{10} + 3052 x^{9} - 7704 x^{8} - 42786 x^{7} - 3072 x^{6} + 330672 x^{5} + 360336 x^{4} - 505624 x^{3} - 727680 x^{2} - 218688 x - 265728 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(16524388251843851540496384000000=2^{30}\cdot 3^{20}\cdot 5^{6}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $120.56$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{4} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} + \frac{1}{4} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{13} + \frac{1}{8} a^{9} - \frac{1}{2} a^{6} - \frac{1}{8} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{374744059249309025431679837208410969920} a^{14} - \frac{1448099647550015252083573250013028261}{46843007406163628178959979651051371240} a^{13} + \frac{1672543169149276990671096874083503371}{187372029624654512715839918604205484960} a^{12} + \frac{171699585486156616017160005953627659}{46843007406163628178959979651051371240} a^{11} - \frac{4428492224676901174244144274222414179}{187372029624654512715839918604205484960} a^{10} - \frac{779976954483899758334610161877226097}{23421503703081814089479989825525685620} a^{9} + \frac{15024171800871544851375738893243426787}{93686014812327256357919959302102742480} a^{8} + \frac{13846415482378447627230279075387538389}{46843007406163628178959979651051371240} a^{7} + \frac{52864098143633265750522975163607887839}{187372029624654512715839918604205484960} a^{6} - \frac{1455962904664501269504432342365182651}{23421503703081814089479989825525685620} a^{5} + \frac{1955356212848349980622729718215707371}{4684300740616362817895997965105137124} a^{4} + \frac{11255166650490310229773110814519330491}{23421503703081814089479989825525685620} a^{3} - \frac{19671013020661997367026111370496653439}{46843007406163628178959979651051371240} a^{2} - \frac{2375905682974111148719054432452598891}{5855375925770453522369997456381421405} a + \frac{1256944634938032809235711315677769411}{5855375925770453522369997456381421405}$

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:  $8$
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:  \( 220936418447 \) (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 $15$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.11.0.1}{11} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ $15$ ${\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.11.0.1}{11} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ $15$ ${\href{/LocalNumberField/43.9.0.1}{9} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.11.0.1}{11} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.13.0.1}{13} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
2.8.21.45$x^{8} + 14 x^{6} + 6 x^{4} + 14$$8$$1$$21$$C_2\wr A_4$$[2, 2, 3, 7/2, 7/2, 15/4]^{3}$
$3$3.3.3.1$x^{3} + 6 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.6.11.13$x^{6} + 6 x^{3} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.6.5$x^{6} + 6 x^{3} + 9 x^{2} + 9$$3$$2$$6$$S_3^2$$[3/2, 3/2]_{2}^{2}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.5.5.2$x^{5} + 5 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.7.0.1$x^{7} - x + 2$$1$$7$$0$$C_7$$[\ ]^{7}$
$7$7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$