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Properties

Label 15.3.20530788642...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{14}\cdot 5^{13}\cdot 23^{6}\cdot 37^{5}$
Root discriminant $89.98$
Ramified primes $2, 5, 23, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T82

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![73858, -84420, -86480, 141555, 5400, -69972, 19630, 5765, -2220, 1650, -1266, 255, -70, 30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 30*x^13 - 70*x^12 + 255*x^11 - 1266*x^10 + 1650*x^9 - 2220*x^8 + 5765*x^7 + 19630*x^6 - 69972*x^5 + 5400*x^4 + 141555*x^3 - 86480*x^2 - 84420*x + 73858)
 
gp: K = bnfinit(x^15 + 30*x^13 - 70*x^12 + 255*x^11 - 1266*x^10 + 1650*x^9 - 2220*x^8 + 5765*x^7 + 19630*x^6 - 69972*x^5 + 5400*x^4 + 141555*x^3 - 86480*x^2 - 84420*x + 73858, 1)
 

Normalized defining polynomial

\( x^{15} + 30 x^{13} - 70 x^{12} + 255 x^{11} - 1266 x^{10} + 1650 x^{9} - 2220 x^{8} + 5765 x^{7} + 19630 x^{6} - 69972 x^{5} + 5400 x^{4} + 141555 x^{3} - 86480 x^{2} - 84420 x + 73858 \)

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:  \(205307886425455460000000000000=2^{14}\cdot 5^{13}\cdot 23^{6}\cdot 37^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.98$
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:  $2, 5, 23, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{12} - \frac{1}{10} a^{11} - \frac{1}{10} a^{10} - \frac{1}{2} a^{8} + \frac{1}{5} a^{7} + \frac{3}{10} a^{6} - \frac{1}{5} a^{5} - \frac{1}{2} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{50} a^{13} + \frac{1}{50} a^{12} + \frac{6}{25} a^{11} - \frac{1}{25} a^{10} + \frac{11}{25} a^{8} + \frac{11}{25} a^{7} - \frac{8}{25} a^{6} - \frac{19}{50} a^{5} + \frac{3}{10} a^{4} + \frac{2}{25} a^{3} - \frac{3}{25} a^{2} - \frac{1}{25} a - \frac{9}{25}$, $\frac{1}{118064934801731135500} a^{14} + \frac{1038265364816053}{118064934801731135500} a^{13} - \frac{4486339569357194261}{118064934801731135500} a^{12} - \frac{14739132275613324003}{118064934801731135500} a^{11} - \frac{6863778427989019401}{29516233700432783875} a^{10} + \frac{224716346152812961}{59032467400865567750} a^{9} + \frac{1897978810837045383}{59032467400865567750} a^{8} - \frac{11678693621724519161}{59032467400865567750} a^{7} + \frac{39675892200259048249}{118064934801731135500} a^{6} + \frac{57570728576635177427}{118064934801731135500} a^{5} - \frac{30026146518035415191}{118064934801731135500} a^{4} - \frac{48033229911753288223}{118064934801731135500} a^{3} + \frac{2925127533224827043}{59032467400865567750} a^{2} - \frac{3585953190881107411}{59032467400865567750} a - \frac{25462721620739137893}{59032467400865567750}$

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:  \( 1723291742.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T82:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48000
The 65 conjugacy class representatives for [F(5)^3]S(3)=F(5)wrS(3) are not computed
Character table for [F(5)^3]S(3)=F(5)wrS(3) is not computed

Intermediate fields

3.3.148.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 30 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ R ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R $15$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.12.27$x^{12} - 18 x^{10} + 171 x^{8} + 116 x^{6} - 313 x^{4} + 190 x^{2} + 877$$6$$2$$12$12T30$[4/3, 4/3]_{3}^{4}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.10.10.6$x^{10} + 10 x^{6} + 10 x^{5} + 75 x^{2} + 50 x + 25$$5$$2$$10$$C_5^2 : C_8$$[5/4, 5/4]_{4}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.8.6.3$x^{8} - 23 x^{4} + 3703$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
37.10.5.1$x^{10} - 2738 x^{6} + 1874161 x^{2} - 11719128733$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$