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Properties

Label 15.15.8604216546...5625.2
Degree $15$
Signature $[15, 0]$
Discriminant $5^{18}\cdot 7^{10}\cdot 41^{8}$
Root discriminant $182.94$
Ramified primes $5, 7, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_5^2 : C_3):C_2$ (as 15T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-21853, 647185, -2437450, -9556485, -7816650, 1717736, 4387410, 922090, -614795, -198850, 29483, 11170, -295, -190, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 190*x^13 - 295*x^12 + 11170*x^11 + 29483*x^10 - 198850*x^9 - 614795*x^8 + 922090*x^7 + 4387410*x^6 + 1717736*x^5 - 7816650*x^4 - 9556485*x^3 - 2437450*x^2 + 647185*x - 21853)
 
gp: K = bnfinit(x^15 - 190*x^13 - 295*x^12 + 11170*x^11 + 29483*x^10 - 198850*x^9 - 614795*x^8 + 922090*x^7 + 4387410*x^6 + 1717736*x^5 - 7816650*x^4 - 9556485*x^3 - 2437450*x^2 + 647185*x - 21853, 1)
 

Normalized defining polynomial

\( x^{15} - 190 x^{13} - 295 x^{12} + 11170 x^{11} + 29483 x^{10} - 198850 x^{9} - 614795 x^{8} + 922090 x^{7} + 4387410 x^{6} + 1717736 x^{5} - 7816650 x^{4} - 9556485 x^{3} - 2437450 x^{2} + 647185 x - 21853 \)

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:  \(8604216546410890678897857666015625=5^{18}\cdot 7^{10}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $182.94$
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:  $5, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7} a^{9} + \frac{1}{7} a^{8} - \frac{1}{7} a^{7} - \frac{2}{7} a^{6} - \frac{3}{7} a^{5} - \frac{3}{7} a^{4} - \frac{2}{7} a^{3} - \frac{1}{7} a^{2} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{287} a^{10} + \frac{138}{287} a^{8} - \frac{8}{287} a^{7} - \frac{64}{287} a^{6} - \frac{17}{41} a^{5} - \frac{1}{7} a^{4} - \frac{1}{7} a^{3} - \frac{2}{7} a^{2} + \frac{1}{7}$, $\frac{1}{287} a^{11} + \frac{15}{287} a^{9} - \frac{131}{287} a^{8} + \frac{59}{287} a^{7} + \frac{127}{287} a^{6} + \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{3}{7} a^{3} + \frac{3}{7} a^{2} - \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{2009} a^{12} - \frac{3}{2009} a^{11} - \frac{2}{2009} a^{10} + \frac{111}{2009} a^{9} - \frac{172}{2009} a^{8} + \frac{660}{2009} a^{7} - \frac{113}{2009} a^{6} - \frac{929}{2009} a^{5} + \frac{18}{49} a^{4} - \frac{20}{49} a^{3} + \frac{23}{49} a^{2} - \frac{4}{49} a - \frac{1}{49}$, $\frac{1}{14063} a^{13} - \frac{2}{14063} a^{12} + \frac{9}{14063} a^{11} - \frac{17}{14063} a^{10} - \frac{712}{14063} a^{9} - \frac{5532}{14063} a^{8} - \frac{4794}{14063} a^{7} + \frac{4495}{14063} a^{6} + \frac{1888}{14063} a^{5} + \frac{152}{343} a^{4} + \frac{129}{343} a^{3} - \frac{9}{343} a^{2} + \frac{93}{343} a - \frac{92}{343}$, $\frac{1}{985794389912272924681485816847} a^{14} - \frac{30721520356410896565312726}{985794389912272924681485816847} a^{13} + \frac{17941022989335774945519235}{140827769987467560668783688121} a^{12} + \frac{188941203322062265590843298}{985794389912272924681485816847} a^{11} - \frac{868424042241759414321304129}{985794389912272924681485816847} a^{10} + \frac{152181371983477981386930288}{10832905383651350820675668317} a^{9} + \frac{260840355765328961812446302486}{985794389912272924681485816847} a^{8} - \frac{40888796839118006523952245092}{985794389912272924681485816847} a^{7} - \frac{268468010404206492154477724733}{985794389912272924681485816847} a^{6} - \frac{54998078922919343744259028140}{140827769987467560668783688121} a^{5} + \frac{6116541440624062731958896516}{24043765607616412797109410167} a^{4} - \frac{719864008017055374672617221}{24043765607616412797109410167} a^{3} + \frac{1606138846000287222426813585}{3434823658230916113872772881} a^{2} + \frac{568767595685881910727505692}{24043765607616412797109410167} a + \frac{779845561047631187459742855}{1849520431355108676700723859}$

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:  \( 1691423909070 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:C_6$ (as 15T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 150
The 10 conjugacy class representatives for $(C_5^2 : C_3):C_2$
Character table for $(C_5^2 : C_3):C_2$

Intermediate fields

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

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

Sibling fields

Degree 15 sibling: data not computed
Degree 25 sibling: data not computed
Degree 30 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{5}$

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
5Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
41Data not computed