↑

Properties

Label 15.15.1120607639...8129.1
Degree $15$
Signature $[15, 0]$
Discriminant $3^{20}\cdot 23^{2}\cdot 59^{2}\cdot 67^{2}\cdot 401^{6}\cdot 967^{2}$
Root discriminant $545.29$
Ramified primes $3, 23, 59, 67, 401, 967$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T80

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-238034595013632, -297543243767040, -148771621883520, -35580044129184, -3036251842164, 372367343943, 96468931584, 3049062912, -812794176, -61956576, 2888136, 361017, -3816, -954, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 954*x^13 - 3816*x^12 + 361017*x^11 + 2888136*x^10 - 61956576*x^9 - 812794176*x^8 + 3049062912*x^7 + 96468931584*x^6 + 372367343943*x^5 - 3036251842164*x^4 - 35580044129184*x^3 - 148771621883520*x^2 - 297543243767040*x - 238034595013632)
 
gp: K = bnfinit(x^15 - 954*x^13 - 3816*x^12 + 361017*x^11 + 2888136*x^10 - 61956576*x^9 - 812794176*x^8 + 3049062912*x^7 + 96468931584*x^6 + 372367343943*x^5 - 3036251842164*x^4 - 35580044129184*x^3 - 148771621883520*x^2 - 297543243767040*x - 238034595013632, 1)
 

Normalized defining polynomial

\( x^{15} - 954 x^{13} - 3816 x^{12} + 361017 x^{11} + 2888136 x^{10} - 61956576 x^{9} - 812794176 x^{8} + 3049062912 x^{7} + 96468931584 x^{6} + 372367343943 x^{5} - 3036251842164 x^{4} - 35580044129184 x^{3} - 148771621883520 x^{2} - 297543243767040 x - 238034595013632 \)

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:  \(112060763989017760404926868179021718858129=3^{20}\cdot 23^{2}\cdot 59^{2}\cdot 67^{2}\cdot 401^{6}\cdot 967^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $545.29$
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:  $3, 23, 59, 67, 401, 967$
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 \(a\))

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{9} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{27} a^{5}$, $\frac{1}{162} a^{6} - \frac{1}{18} a^{4} - \frac{1}{2} a$, $\frac{1}{486} a^{7} - \frac{1}{54} a^{5} - \frac{1}{27} a^{4} - \frac{1}{6} a^{2} + \frac{1}{3} a$, $\frac{1}{1458} a^{8} - \frac{1}{81} a^{5} - \frac{1}{18} a^{4} - \frac{1}{18} a^{3} + \frac{1}{9} a^{2} - \frac{1}{2} a$, $\frac{1}{4374} a^{9} + \frac{1}{486} a^{6} - \frac{1}{54} a^{5} + \frac{1}{27} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a$, $\frac{1}{4374} a^{10} - \frac{1}{54} a^{5} + \frac{1}{54} a^{4} - \frac{1}{18} a^{3} + \frac{1}{6} a$, $\frac{1}{419904} a^{11} - \frac{1}{8748} a^{10} + \frac{1}{69984} a^{9} - \frac{1}{5832} a^{8} - \frac{5}{15552} a^{7} + \frac{1}{1944} a^{6} - \frac{1}{81} a^{5} - \frac{1}{18} a^{4} - \frac{1}{54} a^{3} - \frac{1}{6} a^{2} - \frac{25}{64} a + \frac{7}{16}$, $\frac{1}{5401645056} a^{12} + \frac{5}{6718464} a^{11} - \frac{16261}{300091392} a^{10} + \frac{3829}{37511424} a^{9} + \frac{17801}{66686976} a^{8} + \frac{16775}{16671744} a^{7} - \frac{1247}{2083968} a^{6} + \frac{1949}{347328} a^{5} - \frac{86}{1809} a^{4} + \frac{2}{201} a^{3} + \frac{77237}{2469888} a^{2} - \frac{35921}{102912} a - \frac{2179}{17152}$, $\frac{1}{345705283584} a^{13} - \frac{1}{43213160448} a^{12} + \frac{7265}{6401949696} a^{11} + \frac{276757}{4801462272} a^{10} - \frac{2641135}{38411698176} a^{9} + \frac{221}{1492992} a^{8} + \frac{102713}{266747904} a^{7} + \frac{8659}{22228992} a^{6} - \frac{83267}{5557248} a^{5} + \frac{943}{231552} a^{4} - \frac{3514081}{474218496} a^{3} - \frac{1117921}{39518208} a^{2} - \frac{1243723}{3293184} a - \frac{1203}{274432}$, $\frac{1}{66375414448128} a^{14} + \frac{5}{5531284537344} a^{13} + \frac{49}{1229174341632} a^{12} - \frac{73459}{115235094528} a^{11} - \frac{234523333}{2458348683264} a^{10} + \frac{11421419}{614587170816} a^{9} + \frac{1097041}{5690621952} a^{8} + \frac{1183115}{2133983232} a^{7} + \frac{40151}{16671744} a^{6} - \frac{735859}{88915968} a^{5} + \frac{237023719}{10116661248} a^{4} + \frac{54775945}{1896873984} a^{3} - \frac{2615393}{105381888} a^{2} + \frac{4769129}{13172736} a - \frac{2010017}{4390912}$

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:  \( 7850049219340000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T80:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 38880
The 48 conjugacy class representatives for [1/2.S(3)^5]D(5)
Character table for [1/2.S(3)^5]D(5) is not computed

Intermediate fields

5.5.160801.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ R $15$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ $15$ $15$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ R

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$3.3.4.1$x^{3} - 3 x^{2} + 21$$3$$1$$4$$C_3$$[2]$
3.6.8.1$x^{6} + 6 x^{5} + 18 x^{2} + 9$$3$$2$$8$$C_3^2:C_4$$[2, 2]^{4}$
3.6.8.1$x^{6} + 6 x^{5} + 18 x^{2} + 9$$3$$2$$8$$C_3^2:C_4$$[2, 2]^{4}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.3.2.1$x^{3} - 23$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.6.0.1$x^{6} - x + 15$$1$$6$$0$$C_6$$[\ ]^{6}$
59Data not computed
$67$67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.3.0.1$x^{3} - x + 16$$1$$3$$0$$C_3$$[\ ]^{3}$
67.4.2.1$x^{4} + 1541 x^{2} + 646416$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
401Data not computed
967Data not computed