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Properties

Label 15.15.7599770770...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{12}\cdot 3^{16}\cdot 5^{16}\cdot 7^{10}$
Root discriminant $114.48$
Ramified primes $2, 3, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T42

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1132, 14680, -6800, -230470, -431210, -130655, 212285, 140785, -12110, -25345, -1901, 1855, 200, -65, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 65*x^13 + 200*x^12 + 1855*x^11 - 1901*x^10 - 25345*x^9 - 12110*x^8 + 140785*x^7 + 212285*x^6 - 130655*x^5 - 431210*x^4 - 230470*x^3 - 6800*x^2 + 14680*x + 1132)
 
gp: K = bnfinit(x^15 - 5*x^14 - 65*x^13 + 200*x^12 + 1855*x^11 - 1901*x^10 - 25345*x^9 - 12110*x^8 + 140785*x^7 + 212285*x^6 - 130655*x^5 - 431210*x^4 - 230470*x^3 - 6800*x^2 + 14680*x + 1132, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 65 x^{13} + 200 x^{12} + 1855 x^{11} - 1901 x^{10} - 25345 x^{9} - 12110 x^{8} + 140785 x^{7} + 212285 x^{6} - 130655 x^{5} - 431210 x^{4} - 230470 x^{3} - 6800 x^{2} + 14680 x + 1132 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7599770770692830625000000000000=2^{12}\cdot 3^{16}\cdot 5^{16}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $114.48$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{12} - \frac{1}{10} a^{11} - \frac{1}{2} a^{9} - \frac{3}{10} a^{8} + \frac{3}{10} a^{7} - \frac{1}{5} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{10} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{627011966662309632718618550} a^{14} + \frac{14681849166855160180970763}{627011966662309632718618550} a^{13} + \frac{44638151644362531874836259}{627011966662309632718618550} a^{12} - \frac{122452988602657256326740074}{313505983331154816359309275} a^{11} - \frac{151878028007934721894130329}{627011966662309632718618550} a^{10} - \frac{156254924317444952906630553}{627011966662309632718618550} a^{9} - \frac{146654445269793201683107449}{627011966662309632718618550} a^{8} - \frac{30797752283611174589933756}{313505983331154816359309275} a^{7} + \frac{65413022910014931647579899}{627011966662309632718618550} a^{6} + \frac{4185815874214738506245127}{627011966662309632718618550} a^{5} + \frac{13727173052667080140252121}{627011966662309632718618550} a^{4} + \frac{39508629272830723967727134}{313505983331154816359309275} a^{3} - \frac{97237122202507982706774253}{313505983331154816359309275} a^{2} - \frac{79701880174043931763446484}{313505983331154816359309275} a - \frac{35925853801579218863773257}{313505983331154816359309275}$

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:  $14$
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:  \( 273937438532 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T42:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1620
The 21 conjugacy class representatives for 1/2[3^4:2]F(5)
Character table for 1/2[3^4:2]F(5) is not computed

Intermediate fields

5.5.2450000.1

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

Sibling fields

Degree 15 sibling: data not computed
Degree 30 siblings: data not computed
Degree 45 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 R R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.12.16.3$x^{12} + 84 x^{11} - 108 x^{10} - 99 x^{9} - 54 x^{8} - 81 x^{7} - 27 x^{6} - 108 x^{5} + 27 x^{3} + 81 x^{2} - 81$$3$$4$$16$12T73$[2, 2]^{12}$
$5$5.5.5.1$x^{5} + 20 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.10.11.1$x^{10} + 20 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.12.10.4$x^{12} - 7 x^{6} + 147$$6$$2$$10$$C_{12}$$[\ ]_{6}^{2}$