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Properties

Label 15.1.12271746993...9375.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,5^{26}\cdot 7^{7}$
Root discriminant $40.36$
Ramified primes $5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{15}$ (as 15T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2601, 3315, -6965, 5150, -2575, 353, 120, 705, -400, 75, -38, 40, -15, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 15*x^12 + 40*x^11 - 38*x^10 + 75*x^9 - 400*x^8 + 705*x^7 + 120*x^6 + 353*x^5 - 2575*x^4 + 5150*x^3 - 6965*x^2 + 3315*x - 2601)
 
gp: K = bnfinit(x^15 - 15*x^12 + 40*x^11 - 38*x^10 + 75*x^9 - 400*x^8 + 705*x^7 + 120*x^6 + 353*x^5 - 2575*x^4 + 5150*x^3 - 6965*x^2 + 3315*x - 2601, 1)
 

Normalized defining polynomial

\( x^{15} - 15 x^{12} + 40 x^{11} - 38 x^{10} + 75 x^{9} - 400 x^{8} + 705 x^{7} + 120 x^{6} + 353 x^{5} - 2575 x^{4} + 5150 x^{3} - 6965 x^{2} + 3315 x - 2601 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1227174699306488037109375=-\,5^{26}\cdot 7^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.36$
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:  $5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{21} a^{9} - \frac{1}{7} a^{8} + \frac{2}{21} a^{7} - \frac{2}{21} a^{6} + \frac{1}{21} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{3} + \frac{2}{21} a^{2} - \frac{1}{3} a - \frac{3}{7}$, $\frac{1}{63} a^{10} - \frac{1}{63} a^{9} + \frac{1}{21} a^{8} + \frac{1}{7} a^{7} + \frac{4}{63} a^{6} - \frac{5}{21} a^{4} + \frac{1}{63} a^{3} + \frac{25}{63} a^{2} - \frac{1}{7} a - \frac{2}{7}$, $\frac{1}{63} a^{11} - \frac{1}{63} a^{9} + \frac{1}{9} a^{7} + \frac{10}{63} a^{6} + \frac{1}{21} a^{5} + \frac{13}{63} a^{4} + \frac{17}{63} a^{3} + \frac{31}{63} a^{2} - \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{3969} a^{12} + \frac{23}{3969} a^{11} - \frac{22}{3969} a^{10} - \frac{17}{3969} a^{9} - \frac{599}{3969} a^{8} + \frac{103}{1323} a^{7} - \frac{556}{3969} a^{6} - \frac{500}{3969} a^{5} + \frac{31}{3969} a^{4} + \frac{188}{3969} a^{3} - \frac{1780}{3969} a^{2} + \frac{314}{1323} a + \frac{199}{441}$, $\frac{1}{3969} a^{13} + \frac{16}{3969} a^{11} - \frac{5}{1323} a^{10} - \frac{82}{3969} a^{9} + \frac{100}{3969} a^{8} + \frac{86}{3969} a^{7} - \frac{104}{1323} a^{6} + \frac{191}{3969} a^{5} - \frac{25}{189} a^{4} - \frac{242}{567} a^{3} + \frac{932}{3969} a^{2} - \frac{514}{1323} a - \frac{104}{441}$, $\frac{1}{2643628777839} a^{14} + \frac{2884810}{155507575167} a^{13} - \frac{11413597}{155507575167} a^{12} + \frac{3899626160}{2643628777839} a^{11} - \frac{2173382752}{377661253977} a^{10} + \frac{381736102}{64478750679} a^{9} + \frac{47293418737}{377661253977} a^{8} + \frac{28996168753}{2643628777839} a^{7} + \frac{155713333169}{2643628777839} a^{6} - \frac{126316975541}{2643628777839} a^{5} + \frac{863930648500}{2643628777839} a^{4} + \frac{164467985131}{377661253977} a^{3} - \frac{1034753844197}{2643628777839} a^{2} - \frac{184610378429}{881209592613} a + \frac{8183585395}{17278619463}$

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:  $7$
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:  \( 6587237.04765 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{15}$ (as 15T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 30
The 9 conjugacy class representatives for $D_{15}$
Character table for $D_{15}$

Intermediate fields

3.1.175.1, 5.1.19140625.1

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

Sibling fields

Galois closure: 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 $15$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R R $15$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
5Data not computed
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$