Properties

Label 15.5.154972454814106259.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,11^{13}\cdot 67^{2}$
Root discriminant $14.00$
Ramified primes $11, 67$
Class number $1$
Class group Trivial
Galois Group 15T44

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -7, 27, -7, -34, 28, 8, -15, 2, -1, 1, 1, -2, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - x^14 - 2*x^13 + x^12 + x^11 - x^10 + 2*x^9 - 15*x^8 + 8*x^7 + 28*x^6 - 34*x^5 - 7*x^4 + 27*x^3 - 7*x^2 - 3*x + 1)
gp: K = bnfinit(x^15 - x^14 - 2*x^13 + x^12 + x^11 - x^10 + 2*x^9 - 15*x^8 + 8*x^7 + 28*x^6 - 34*x^5 - 7*x^4 + 27*x^3 - 7*x^2 - 3*x + 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut x^{14} \) \(\mathstrut -\mathstrut 2 x^{13} \) \(\mathstrut +\mathstrut x^{12} \) \(\mathstrut +\mathstrut x^{11} \) \(\mathstrut -\mathstrut x^{10} \) \(\mathstrut +\mathstrut 2 x^{9} \) \(\mathstrut -\mathstrut 15 x^{8} \) \(\mathstrut +\mathstrut 8 x^{7} \) \(\mathstrut +\mathstrut 28 x^{6} \) \(\mathstrut -\mathstrut 34 x^{5} \) \(\mathstrut -\mathstrut 7 x^{4} \) \(\mathstrut +\mathstrut 27 x^{3} \) \(\mathstrut -\mathstrut 7 x^{2} \) \(\mathstrut -\mathstrut 3 x \) \(\mathstrut +\mathstrut 1 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[5, 5]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-154972454814106259=-\,11^{13}\cdot 67^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $14.00$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $11, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1375967} a^{14} + \frac{175735}{1375967} a^{13} - \frac{613357}{1375967} a^{12} + \frac{221128}{1375967} a^{11} + \frac{90195}{1375967} a^{10} - \frac{631321}{1375967} a^{9} - \frac{232077}{1375967} a^{8} - \frac{621807}{1375967} a^{7} - \frac{79672}{1375967} a^{6} + \frac{601628}{1375967} a^{5} - \frac{230139}{1375967} a^{4} + \frac{90720}{1375967} a^{3} - \frac{559682}{1375967} a^{2} + \frac{597135}{1375967} a - \frac{6898}{1375967}$

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:  $9$
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:  \( \frac{9871118}{1375967} a^{14} - \frac{5442576}{1375967} a^{13} - \frac{22352934}{1375967} a^{12} + \frac{195017}{1375967} a^{11} + \frac{10168561}{1375967} a^{10} - \frac{5609891}{1375967} a^{9} + \frac{16901521}{1375967} a^{8} - \frac{140513656}{1375967} a^{7} + \frac{15248762}{1375967} a^{6} + \frac{286186857}{1375967} a^{5} - \frac{211351518}{1375967} a^{4} - \frac{166885954}{1375967} a^{3} + \frac{198831449}{1375967} a^{2} + \frac{18484429}{1375967} a - \frac{24636408}{1375967} \),  \( \frac{2954835}{1375967} a^{14} - \frac{1377570}{1375967} a^{13} - \frac{6917210}{1375967} a^{12} - \frac{439608}{1375967} a^{11} + \frac{3046529}{1375967} a^{10} - \frac{1390323}{1375967} a^{9} + \frac{5191165}{1375967} a^{8} - \frac{41374953}{1375967} a^{7} + \frac{707811}{1375967} a^{6} + \frac{87320377}{1375967} a^{5} - \frac{57031774}{1375967} a^{4} - \frac{55546486}{1375967} a^{3} + \frac{57685609}{1375967} a^{2} + \frac{8146219}{1375967} a - \frac{7132494}{1375967} \),  \( \frac{1450324}{1375967} a^{14} - \frac{431204}{1375967} a^{13} - \frac{3712168}{1375967} a^{12} - \frac{390954}{1375967} a^{11} + \frac{1542424}{1375967} a^{10} - \frac{645425}{1375967} a^{9} + \frac{2204592}{1375967} a^{8} - \frac{19723503}{1375967} a^{7} - \frac{3384903}{1375967} a^{6} + \frac{46596970}{1375967} a^{5} - \frac{25687417}{1375967} a^{4} - \frac{33722369}{1375967} a^{3} + \frac{30118715}{1375967} a^{2} + \frac{5591940}{1375967} a - \frac{3806796}{1375967} \),  \( a \),  \( \frac{5494597}{1375967} a^{14} - \frac{2846191}{1375967} a^{13} - \frac{12822567}{1375967} a^{12} + \frac{113142}{1375967} a^{11} + \frac{5893959}{1375967} a^{10} - \frac{3138561}{1375967} a^{9} + \frac{9205248}{1375967} a^{8} - \frac{77583185}{1375967} a^{7} + \frac{5248701}{1375967} a^{6} + \frac{164217103}{1375967} a^{5} - \frac{117463343}{1375967} a^{4} - \frac{100794874}{1375967} a^{3} + \frac{115621493}{1375967} a^{2} + \frac{10484392}{1375967} a - \frac{15854728}{1375967} \),  \( \frac{13996050}{1375967} a^{14} - \frac{8460232}{1375967} a^{13} - \frac{31332111}{1375967} a^{12} + \frac{1378211}{1375967} a^{11} + \frac{14821072}{1375967} a^{10} - \frac{8036830}{1375967} a^{9} + \frac{24834469}{1375967} a^{8} - \frac{200082968}{1375967} a^{7} + \frac{33016177}{1375967} a^{6} + \frac{404291653}{1375967} a^{5} - \frac{312370017}{1375967} a^{4} - \frac{224334716}{1375967} a^{3} + \frac{287797026}{1375967} a^{2} + \frac{20588044}{1375967} a - \frac{35803487}{1375967} \),  \( \frac{5483797}{1375967} a^{14} - \frac{3325698}{1375967} a^{13} - \frac{12472105}{1375967} a^{12} + \frac{609454}{1375967} a^{11} + \frac{5972595}{1375967} a^{10} - \frac{2788246}{1375967} a^{9} + \frac{10000941}{1375967} a^{8} - \frac{78162512}{1375967} a^{7} + \frac{12606761}{1375967} a^{6} + \frac{161198943}{1375967} a^{5} - \frac{122462413}{1375967} a^{4} - \frac{88498667}{1375967} a^{3} + \frac{114188095}{1375967} a^{2} + \frac{7831787}{1375967} a - \frac{12906612}{1375967} \),  \( \frac{10978394}{1375967} a^{14} - \frac{6171889}{1375967} a^{13} - \frac{25294672}{1375967} a^{12} + \frac{722596}{1375967} a^{11} + \frac{11866554}{1375967} a^{10} - \frac{5926807}{1375967} a^{9} + \frac{19206189}{1375967} a^{8} - \frac{155745697}{1375967} a^{7} + \frac{17855462}{1375967} a^{6} + \frac{325416046}{1375967} a^{5} - \frac{239925756}{1375967} a^{4} - \frac{189293541}{1375967} a^{3} + \frac{229809588}{1375967} a^{2} + \frac{18316179}{1375967} a - \frac{28761340}{1375967} \),  \( \frac{17550747}{1375967} a^{14} - \frac{10049003}{1375967} a^{13} - \frac{39734123}{1375967} a^{12} + \frac{1252205}{1375967} a^{11} + \frac{18022284}{1375967} a^{10} - \frac{10259379}{1375967} a^{9} + \frac{30689452}{1375967} a^{8} - \frac{249594030}{1375967} a^{7} + \frac{32980535}{1375967} a^{6} + \frac{511898243}{1375967} a^{5} - \frac{386785103}{1375967} a^{4} - \frac{288151214}{1375967} a^{3} + \frac{363226421}{1375967} a^{2} + \frac{23216490}{1375967} a - \frac{43251288}{1375967} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 492.79424719 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

15T44:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 2430
The 39 conjugacy class representatives for [3^5:2]5
Character table for [3^5:2]5 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\)

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

Sibling fields

Degree 15 siblings: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{3}$ $15$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ $15$ $15$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ $15$ $15$ $15$

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
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
67Data not computed