Properties

Label 15.3.16680932154277329.1
Degree $15$
Signature $[3, 6]$
Discriminant $3\cdot 47\cdot 103\cdot 11057\cdot 103878739$
Root discriminant $12.06$
Ramified primes $3, 47, 103, 11057, 103878739$
Class number $1$
Class group Trivial
Galois Group 15T104

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 17, -30, 34, -20, -4, 25, -41, 44, -35, 23, -14, 6, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 6*x^13 - 14*x^12 + 23*x^11 - 35*x^10 + 44*x^9 - 41*x^8 + 25*x^7 - 4*x^6 - 20*x^5 + 34*x^4 - 30*x^3 + 17*x^2 - 6*x + 1)
gp: K = bnfinit(x^15 - 2*x^14 + 6*x^13 - 14*x^12 + 23*x^11 - 35*x^10 + 44*x^9 - 41*x^8 + 25*x^7 - 4*x^6 - 20*x^5 + 34*x^4 - 30*x^3 + 17*x^2 - 6*x + 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 2 x^{14} \) \(\mathstrut +\mathstrut 6 x^{13} \) \(\mathstrut -\mathstrut 14 x^{12} \) \(\mathstrut +\mathstrut 23 x^{11} \) \(\mathstrut -\mathstrut 35 x^{10} \) \(\mathstrut +\mathstrut 44 x^{9} \) \(\mathstrut -\mathstrut 41 x^{8} \) \(\mathstrut +\mathstrut 25 x^{7} \) \(\mathstrut -\mathstrut 4 x^{6} \) \(\mathstrut -\mathstrut 20 x^{5} \) \(\mathstrut +\mathstrut 34 x^{4} \) \(\mathstrut -\mathstrut 30 x^{3} \) \(\mathstrut +\mathstrut 17 x^{2} \) \(\mathstrut -\mathstrut 6 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(16680932154277329=3\cdot 47\cdot 103\cdot 11057\cdot 103878739\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $12.06$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 47, 103, 11057, 103878739$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{229} a^{14} - \frac{80}{229} a^{13} + \frac{63}{229} a^{12} + \frac{110}{229} a^{11} - \frac{84}{229} a^{10} + \frac{105}{229} a^{9} + \frac{98}{229} a^{8} + \frac{101}{229} a^{7} - \frac{67}{229} a^{6} - \frac{45}{229} a^{5} + \frac{55}{229} a^{4} + \frac{95}{229} a^{3} - \frac{112}{229} a^{2} + \frac{51}{229} a - \frac{91}{229}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( a \),  \( \frac{837}{229} a^{14} - \frac{1237}{229} a^{13} + \frac{4412}{229} a^{12} - \frac{9377}{229} a^{11} + \frac{14422}{229} a^{10} - \frac{21806}{229} a^{9} + \frac{25234}{229} a^{8} - \frac{21032}{229} a^{7} + \frac{9644}{229} a^{6} + \frac{2181}{229} a^{5} - \frac{16024}{229} a^{4} + \frac{19975}{229} a^{3} - \frac{14739}{229} a^{2} + \frac{6505}{229} a - \frac{1284}{229} \),  \( \frac{93}{229} a^{14} - \frac{341}{229} a^{13} + \frac{592}{229} a^{12} - \frac{1907}{229} a^{11} + \frac{2951}{229} a^{10} - \frac{4204}{229} a^{9} + \frac{5679}{229} a^{8} - \frac{4805}{229} a^{7} + \frac{2700}{229} a^{6} + \frac{166}{229} a^{5} - \frac{2671}{229} a^{4} + \frac{4713}{229} a^{3} - \frac{3546}{229} a^{2} + \frac{1995}{229} a - \frac{448}{229} \),  \( \frac{223}{229} a^{14} + \frac{22}{229} a^{13} + \frac{767}{229} a^{12} - \frac{889}{229} a^{11} + \frac{504}{229} a^{10} - \frac{1317}{229} a^{9} - \frac{130}{229} a^{8} + \frac{1684}{229} a^{7} - \frac{2117}{229} a^{6} + \frac{1644}{229} a^{5} - \frac{2162}{229} a^{4} - \frac{570}{229} a^{3} + \frac{1817}{229} a^{2} - \frac{993}{229} a + \frac{546}{229} \),  \( \frac{592}{229} a^{14} - \frac{644}{229} a^{13} + \frac{2946}{229} a^{12} - \frac{5641}{229} a^{11} + \frac{8438}{229} a^{10} - \frac{13181}{229} a^{9} + \frac{14277}{229} a^{8} - \frac{11656}{229} a^{7} + \frac{4991}{229} a^{6} + \frac{1527}{229} a^{5} - \frac{10034}{229} a^{4} + \frac{11127}{229} a^{3} - \frac{8138}{229} a^{2} + \frac{3857}{229} a - \frac{744}{229} \),  \( \frac{52}{229} a^{14} - \frac{38}{229} a^{13} + \frac{299}{229} a^{12} - \frac{463}{229} a^{11} + \frac{899}{229} a^{10} - \frac{1410}{229} a^{9} + \frac{1661}{229} a^{8} - \frac{1847}{229} a^{7} + \frac{1096}{229} a^{6} - \frac{279}{229} a^{5} - \frac{1033}{229} a^{4} + \frac{1505}{229} a^{3} - \frac{1931}{229} a^{2} + \frac{820}{229} a - \frac{152}{229} \),  \( \frac{769}{229} a^{14} - \frac{1293}{229} a^{13} + \frac{4021}{229} a^{12} - \frac{9300}{229} a^{11} + \frac{13951}{229} a^{10} - \frac{20931}{229} a^{9} + \frac{25211}{229} a^{8} - \frac{20572}{229} a^{7} + \frac{9849}{229} a^{6} + \frac{1577}{229} a^{5} - \frac{14726}{229} a^{4} + \frac{20156}{229} a^{3} - \frac{13993}{229} a^{2} + \frac{6472}{229} a - \frac{1508}{229} \),  \( \frac{1431}{229} a^{14} - \frac{1812}{229} a^{13} + \frac{7255}{229} a^{12} - \frac{14798}{229} a^{11} + \frac{22005}{229} a^{10} - \frac{34090}{229} a^{9} + \frac{38104}{229} a^{8} - \frac{30425}{229} a^{7} + \frac{13127}{229} a^{6} + \frac{4534}{229} a^{5} - \frac{26177}{229} a^{4} + \frac{30147}{229} a^{3} - \frac{20353}{229} a^{2} + \frac{8632}{229} a - \frac{1523}{229} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 124.641970008 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

15T104:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 1307674368000
The 176 conjugacy class representatives for S15 are not computed
Character table for S15 is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 30 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 $15$ R $15$ ${\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ $15$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.11.0.1}{11} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.11.0.1}{11} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.13.0.1}{13} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.9.0.1$x^{9} - x^{3} + x^{2} + 1$$1$$9$$0$$C_9$$[\ ]^{9}$
$47$47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.13.0.1$x^{13} - x + 32$$1$$13$$0$$C_{13}$$[\ ]^{13}$
$103$103.2.1.2$x^{2} + 206$$2$$1$$1$$C_2$$[\ ]_{2}$
103.5.0.1$x^{5} - x + 18$$1$$5$$0$$C_5$$[\ ]^{5}$
103.8.0.1$x^{8} - x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
11057Data not computed
103878739Data not computed