Properties

Label 15.3.4864965285308625.1
Degree $15$
Signature $[3, 6]$
Discriminant $3^{9}\cdot 5^{3}\cdot 7^{11}$
Root discriminant $11.11$
Ramified primes $3, 5, 7$
Class number $1$
Class group Trivial
Galois Group $S_5 \times C_3$ (as 15T24)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 5, -11, 16, -15, 18, -21, 31, -28, 24, -17, 14, -12, 6, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 + 6*x^13 - 12*x^12 + 14*x^11 - 17*x^10 + 24*x^9 - 28*x^8 + 31*x^7 - 21*x^6 + 18*x^5 - 15*x^4 + 16*x^3 - 11*x^2 + 5*x - 1)
gp: K = bnfinit(x^15 - 3*x^14 + 6*x^13 - 12*x^12 + 14*x^11 - 17*x^10 + 24*x^9 - 28*x^8 + 31*x^7 - 21*x^6 + 18*x^5 - 15*x^4 + 16*x^3 - 11*x^2 + 5*x - 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 3 x^{14} \) \(\mathstrut +\mathstrut 6 x^{13} \) \(\mathstrut -\mathstrut 12 x^{12} \) \(\mathstrut +\mathstrut 14 x^{11} \) \(\mathstrut -\mathstrut 17 x^{10} \) \(\mathstrut +\mathstrut 24 x^{9} \) \(\mathstrut -\mathstrut 28 x^{8} \) \(\mathstrut +\mathstrut 31 x^{7} \) \(\mathstrut -\mathstrut 21 x^{6} \) \(\mathstrut +\mathstrut 18 x^{5} \) \(\mathstrut -\mathstrut 15 x^{4} \) \(\mathstrut +\mathstrut 16 x^{3} \) \(\mathstrut -\mathstrut 11 x^{2} \) \(\mathstrut +\mathstrut 5 x \) \(\mathstrut -\mathstrut 1 \)

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:  \(4864965285308625=3^{9}\cdot 5^{3}\cdot 7^{11}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.11$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{81997} a^{14} - \frac{36770}{81997} a^{13} + \frac{38057}{81997} a^{12} + \frac{222}{491} a^{11} + \frac{18384}{81997} a^{10} - \frac{23274}{81997} a^{9} - \frac{5510}{81997} a^{8} - \frac{28445}{81997} a^{7} - \frac{34389}{81997} a^{6} - \frac{13398}{81997} a^{5} - \frac{33692}{81997} a^{4} + \frac{25070}{81997} a^{3} - \frac{20397}{81997} a^{2} - \frac{8074}{81997} a + \frac{27623}{81997}$

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:  \( a \),  \( \frac{1892}{81997} a^{14} + \frac{46613}{81997} a^{13} - \frac{71519}{81997} a^{12} + \frac{710}{491} a^{11} - \frac{312188}{81997} a^{10} + \frac{79978}{81997} a^{9} - \frac{339289}{81997} a^{8} + \frac{546071}{81997} a^{7} - \frac{204361}{81997} a^{6} + \frac{562036}{81997} a^{5} + \frac{48402}{81997} a^{4} + \frac{448159}{81997} a^{3} - \frac{134531}{81997} a^{2} + \frac{221425}{81997} a - \frac{51370}{81997} \),  \( \frac{29802}{81997} a^{14} - \frac{11632}{81997} a^{13} - \frac{7790}{81997} a^{12} - \frac{181}{491} a^{11} - \frac{269977}{81997} a^{10} + \frac{82872}{81997} a^{9} - \frac{51026}{81997} a^{8} + \frac{459078}{81997} a^{7} - \frac{226466}{81997} a^{6} + \frac{694170}{81997} a^{5} - \frac{35719}{81997} a^{4} + \frac{143470}{81997} a^{3} - \frac{191627}{81997} a^{2} + \frac{203841}{81997} a - \frac{193228}{81997} \),  \( \frac{149}{81997} a^{14} - \frac{66928}{81997} a^{13} + \frac{94697}{81997} a^{12} - \frac{801}{491} a^{11} + \frac{361303}{81997} a^{10} - \frac{23952}{81997} a^{9} + \frac{326968}{81997} a^{8} - \frac{712434}{81997} a^{7} + \frac{123847}{81997} a^{6} - \frac{438359}{81997} a^{5} - \frac{100288}{81997} a^{4} - \frac{446417}{81997} a^{3} + \frac{240727}{81997} a^{2} - \frac{55068}{81997} a - \frac{66020}{81997} \),  \( \frac{3014}{81997} a^{14} - \frac{46833}{81997} a^{13} + \frac{71992}{81997} a^{12} - \frac{616}{491} a^{11} + \frac{225395}{81997} a^{10} - \frac{40401}{81997} a^{9} + \frac{202245}{81997} a^{8} - \frac{456350}{81997} a^{7} + \frac{159756}{81997} a^{6} - \frac{203042}{81997} a^{5} - \frac{117399}{81997} a^{4} - \frac{286245}{81997} a^{3} + \frac{185186}{81997} a^{2} - \frac{63924}{81997} a - \frac{53230}{81997} \),  \( \frac{129030}{81997} a^{14} - \frac{250674}{81997} a^{13} + \frac{432353}{81997} a^{12} - \frac{5681}{491} a^{11} + \frac{570286}{81997} a^{10} - \frac{1052053}{81997} a^{9} + \frac{1680627}{81997} a^{8} - \frac{1220588}{81997} a^{7} + \frac{1612928}{81997} a^{6} - \frac{247180}{81997} a^{5} + \frac{1104147}{81997} a^{4} - \frac{573529}{81997} a^{3} + \frac{770772}{81997} a^{2} - \frac{180329}{81997} a + \frac{32091}{81997} \),  \( \frac{47103}{81997} a^{14} - \frac{200670}{81997} a^{13} + \frac{390442}{81997} a^{12} - \frac{4380}{491} a^{11} + \frac{1037196}{81997} a^{10} - \frac{959296}{81997} a^{9} + \frac{1458921}{81997} a^{8} - \frac{1981783}{81997} a^{7} + \frac{1829602}{81997} a^{6} - \frac{1431031}{81997} a^{5} + \frac{795632}{81997} a^{4} - \frac{1032548}{81997} a^{3} + \frac{900925}{81997} a^{2} - \frac{745509}{81997} a + \frac{243764}{81997} \),  \( \frac{95611}{81997} a^{14} - \frac{241086}{81997} a^{13} + \frac{460937}{81997} a^{12} - \frac{5689}{491} a^{11} + \frac{926899}{81997} a^{10} - \frac{1245783}{81997} a^{9} + \frac{1818049}{81997} a^{8} - \frac{1864330}{81997} a^{7} + \frac{2162946}{81997} a^{6} - \frac{1187002}{81997} a^{5} + \frac{1238285}{81997} a^{4} - \frac{870501}{81997} a^{3} + \frac{1023045}{81997} a^{2} - \frac{617435}{81997} a + \frac{185274}{81997} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 50.4138149085 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_3\times S_5$ (as 15T24):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 360
The 21 conjugacy class representatives for $S_5 \times C_3$
Character table for $S_5 \times C_3$ is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 5.1.46305.1

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

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 sibling: 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 $15$ R R R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{5}$

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$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.9.9.1$x^{9} + 54 x^{5} + 27 x^{3} + 189$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$