Properties

Label 15.3.146928604515779584.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{10}\cdot 3461^{4}$
Root discriminant $13.95$
Ramified primes $2, 3461$
Class number $1$
Class group Trivial
Galois Group 15T28

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -1, 7, -1, -1, 0, 1, -1, 4, -5, -1, 9, -3, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 3*x^13 + 9*x^12 - x^11 - 5*x^10 + 4*x^9 - x^8 + x^7 - x^5 - x^4 + 7*x^3 - x^2 + 1)
gp: K = bnfinit(x^15 - 2*x^14 - 3*x^13 + 9*x^12 - x^11 - 5*x^10 + 4*x^9 - x^8 + x^7 - x^5 - x^4 + 7*x^3 - x^2 + 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 2 x^{14} \) \(\mathstrut -\mathstrut 3 x^{13} \) \(\mathstrut +\mathstrut 9 x^{12} \) \(\mathstrut -\mathstrut x^{11} \) \(\mathstrut -\mathstrut 5 x^{10} \) \(\mathstrut +\mathstrut 4 x^{9} \) \(\mathstrut -\mathstrut x^{8} \) \(\mathstrut +\mathstrut x^{7} \) \(\mathstrut -\mathstrut x^{5} \) \(\mathstrut -\mathstrut x^{4} \) \(\mathstrut +\mathstrut 7 x^{3} \) \(\mathstrut -\mathstrut x^{2} \) \(\mathstrut +\mathstrut 1 \)

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:  \(146928604515779584=2^{10}\cdot 3461^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $13.95$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3461$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{47524} a^{14} + \frac{2413}{47524} a^{13} + \frac{2851}{23762} a^{12} - \frac{11621}{47524} a^{11} - \frac{897}{23762} a^{10} - \frac{7831}{47524} a^{9} - \frac{21071}{47524} a^{8} - \frac{3006}{11881} a^{7} - \frac{795}{47524} a^{6} + \frac{4797}{47524} a^{5} + \frac{3165}{11881} a^{4} + \frac{15967}{47524} a^{3} - \frac{2707}{23762} a^{2} + \frac{18051}{47524} a - \frac{10105}{47524}$

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:  \( a \),  \( \frac{2319}{11881} a^{14} - \frac{204}{11881} a^{13} - \frac{61865}{47524} a^{12} + \frac{23679}{47524} a^{11} + \frac{146709}{47524} a^{10} - \frac{5921}{11881} a^{9} - \frac{71551}{47524} a^{8} - \frac{7677}{47524} a^{7} + \frac{9831}{11881} a^{6} + \frac{26389}{47524} a^{5} - \frac{9525}{47524} a^{4} - \frac{5604}{11881} a^{3} + \frac{24485}{47524} a^{2} + \frac{120953}{47524} a + \frac{18991}{47524} \),  \( \frac{3566}{11881} a^{14} - \frac{8967}{11881} a^{13} - \frac{6940}{11881} a^{12} + \frac{36085}{11881} a^{11} - \frac{17307}{11881} a^{10} - \frac{21873}{23762} a^{9} + \frac{8139}{11881} a^{8} - \frac{10936}{11881} a^{7} + \frac{44821}{23762} a^{6} - \frac{2538}{11881} a^{5} - \frac{14121}{11881} a^{4} - \frac{26503}{23762} a^{3} + \frac{35944}{11881} a^{2} - \frac{13273}{11881} a + \frac{13167}{23762} \),  \( \frac{2262}{11881} a^{14} - \frac{7054}{11881} a^{13} - \frac{7487}{47524} a^{12} + \frac{106971}{47524} a^{11} - \frac{85833}{47524} a^{10} - \frac{10183}{23762} a^{9} + \frac{27761}{47524} a^{8} - \frac{46359}{47524} a^{7} + \frac{27125}{23762} a^{6} - \frac{21799}{47524} a^{5} + \frac{26721}{47524} a^{4} - \frac{13653}{23762} a^{3} + \frac{70777}{47524} a^{2} - \frac{73945}{47524} a - \frac{5745}{47524} \),  \( \frac{6459}{47524} a^{14} - \frac{26067}{47524} a^{13} + \frac{9999}{47524} a^{12} + \frac{43643}{23762} a^{11} - \frac{122281}{47524} a^{10} + \frac{8869}{47524} a^{9} + \frac{23395}{23762} a^{8} - \frac{20681}{47524} a^{7} + \frac{21449}{47524} a^{6} - \frac{30615}{23762} a^{5} + \frac{17779}{47524} a^{4} - \frac{19989}{47524} a^{3} + \frac{44281}{47524} a^{2} - \frac{22973}{11881} a + \frac{4475}{11881} \),  \( \frac{4779}{11881} a^{14} - \frac{21329}{23762} a^{13} - \frac{22193}{23762} a^{12} + \frac{42559}{11881} a^{11} - \frac{26531}{23762} a^{10} - \frac{11080}{11881} a^{9} + \frac{5047}{11881} a^{8} - \frac{479}{23762} a^{7} + \frac{14496}{11881} a^{6} - \frac{17348}{11881} a^{5} + \frac{20057}{23762} a^{4} - \frac{17251}{11881} a^{3} + \frac{27074}{11881} a^{2} - \frac{2212}{11881} a - \frac{2941}{23762} \),  \( \frac{2263}{11881} a^{14} - \frac{30445}{47524} a^{13} - \frac{8441}{47524} a^{12} + \frac{131773}{47524} a^{11} - \frac{52445}{23762} a^{10} - \frac{87333}{47524} a^{9} + \frac{109811}{47524} a^{8} - \frac{17525}{23762} a^{7} + \frac{15427}{47524} a^{6} - \frac{26373}{47524} a^{5} - \frac{2903}{23762} a^{4} + \frac{48443}{47524} a^{3} + \frac{49121}{47524} a^{2} - \frac{73027}{47524} a - \frac{5261}{23762} \),  \( \frac{1879}{23762} a^{14} - \frac{20911}{47524} a^{13} + \frac{6673}{47524} a^{12} + \frac{86005}{47524} a^{11} - \frac{22123}{11881} a^{10} - \frac{70947}{47524} a^{9} + \frac{49571}{47524} a^{8} + \frac{2283}{11881} a^{7} - \frac{5479}{47524} a^{6} - \frac{67637}{47524} a^{5} - \frac{10692}{11881} a^{4} - \frac{30707}{47524} a^{3} + \frac{77627}{47524} a^{2} - \frac{40495}{47524} a - \frac{1457}{23762} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 470.017972215 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

15T28:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 720
The 11 conjugacy class representatives for S_6(15)
Character table for S_6(15)

Intermediate fields

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

Sibling fields

Degree 6 siblings: 6.2.165830644724.1, 6.2.55376.1
Degree 10 sibling: 10.2.2653290315584.1
Degree 12 siblings: Deg 12, Deg 12
Degree 15 sibling: Deg 15
Degree 20 siblings: 20.4.7039949498771842313261056.1, Deg 20, Deg 20
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Degree 45 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
3461Data not computed