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Properties

Label 15.15.3710364839...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{12}\cdot 5^{4}\cdot 13^{13}\cdot 2187542681^{2}$
Root discriminant $434.47$
Ramified primes $2, 5, 13, 2187542681$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T59

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1612488125, -1309842980, 1160684561, 988858909, -260610108, -249268798, 17147018, 25291850, -142563, -1207192, -15049, 28495, 333, -306, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 306*x^13 + 333*x^12 + 28495*x^11 - 15049*x^10 - 1207192*x^9 - 142563*x^8 + 25291850*x^7 + 17147018*x^6 - 249268798*x^5 - 260610108*x^4 + 988858909*x^3 + 1160684561*x^2 - 1309842980*x - 1612488125)
 
gp: K = bnfinit(x^15 - x^14 - 306*x^13 + 333*x^12 + 28495*x^11 - 15049*x^10 - 1207192*x^9 - 142563*x^8 + 25291850*x^7 + 17147018*x^6 - 249268798*x^5 - 260610108*x^4 + 988858909*x^3 + 1160684561*x^2 - 1309842980*x - 1612488125, 1)
 

Normalized defining polynomial

\( x^{15} - x^{14} - 306 x^{13} + 333 x^{12} + 28495 x^{11} - 15049 x^{10} - 1207192 x^{9} - 142563 x^{8} + 25291850 x^{7} + 17147018 x^{6} - 249268798 x^{5} - 260610108 x^{4} + 988858909 x^{3} + 1160684561 x^{2} - 1309842980 x - 1612488125 \)

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:  \(3710364839706702058501959007886164480000=2^{12}\cdot 5^{4}\cdot 13^{13}\cdot 2187542681^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $434.47$
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:  $2, 5, 13, 2187542681$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{20} a^{12} - \frac{1}{10} a^{9} + \frac{3}{20} a^{8} + \frac{1}{10} a^{7} - \frac{1}{10} a^{6} - \frac{2}{5} a^{5} - \frac{1}{2} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{4}$, $\frac{1}{20} a^{13} - \frac{1}{10} a^{10} + \frac{3}{20} a^{9} + \frac{1}{10} a^{8} - \frac{1}{10} a^{7} + \frac{1}{10} a^{6} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4219529971389322678383062557871088439486205123732780} a^{14} - \frac{101477857039587584220772342023307160143828334689}{7961377304508155996949174637492619697143783252326} a^{13} - \frac{1050023126975579176740176209484034200354085724681}{2109764985694661339191531278935544219743102561866390} a^{12} - \frac{395839309513127185107644613022173856570392370306881}{2109764985694661339191531278935544219743102561866390} a^{11} - \frac{415945299336105366905790788966190243625621371223097}{4219529971389322678383062557871088439486205123732780} a^{10} - \frac{18769222710229166786388847735083809008861819199061}{1054882492847330669595765639467772109871551280933195} a^{9} + \frac{3713057112597238990868883399630110916417292213467}{39806886522540779984745873187463098485718916261630} a^{8} - \frac{494109099968879917516111071272093917339455815749481}{2109764985694661339191531278935544219743102561866390} a^{7} + \frac{227443935727190452307177552448338371209186870841481}{1054882492847330669595765639467772109871551280933195} a^{6} - \frac{210491277228813596860223881281246696777407428223511}{1054882492847330669595765639467772109871551280933195} a^{5} + \frac{971079039732823269114419612485699943283048104691381}{2109764985694661339191531278935544219743102561866390} a^{4} + \frac{23008892325918154483881925263849245775779347912331}{1054882492847330669595765639467772109871551280933195} a^{3} - \frac{356040651221232725405047735631053144001153738479449}{4219529971389322678383062557871088439486205123732780} a^{2} - \frac{279933880314451048301704517967177326067365181476647}{1054882492847330669595765639467772109871551280933195} a + \frac{58716372279808475538686075277703225912788772599880}{210976498569466133919153127893554421974310256186639}$

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:  \( 1788919114380000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T59:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 6000
The 28 conjugacy class representatives for [D(5)^3:2]3
Character table for [D(5)^3:2]3 is not computed

Intermediate fields

3.3.169.1

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

Sibling fields

Degree 20 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: 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 $15$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.12.12$x^{12} + 66 x^{10} - 93 x^{8} - 68 x^{6} - 41 x^{4} + 66 x^{2} - 123$$2$$6$$12$12T29$[2, 2, 2]^{6}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.4.0.1$x^{4} + x^{2} - 2 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$
2187542681Data not computed