Properties

Label 15.3.334095024862954369.1
Degree $15$
Signature $[3, 6]$
Discriminant $3.341\times 10^{17}$
Root discriminant $14.73$
Ramified primes $7, 17$
Class number $1$
Class group trivial
Galois group $D_5\times C_3$ (as 15T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 + 3*x^13 - 5*x^12 + x^11 - 19*x^10 + 16*x^9 - 26*x^8 + 55*x^7 - 78*x^6 + 88*x^5 - 54*x^4 + 39*x^3 - 52*x^2 + 21*x - 1)
 
gp: K = bnfinit(x^15 - 2*x^14 + 3*x^13 - 5*x^12 + x^11 - 19*x^10 + 16*x^9 - 26*x^8 + 55*x^7 - 78*x^6 + 88*x^5 - 54*x^4 + 39*x^3 - 52*x^2 + 21*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 21, -52, 39, -54, 88, -78, 55, -26, 16, -19, 1, -5, 3, -2, 1]);
 

\( x^{15} - 2 x^{14} + 3 x^{13} - 5 x^{12} + x^{11} - 19 x^{10} + 16 x^{9} - 26 x^{8} + 55 x^{7} - 78 x^{6} + 88 x^{5} - 54 x^{4} + 39 x^{3} - 52 x^{2} + 21 x - 1 \)

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

Invariants

Degree:  $15$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[3, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(334095024862954369\)\(\medspace = 7^{12}\cdot 17^{6}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $14.73$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $7, 17$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $3$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{7} + \frac{1}{7} a^{6} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{2}{7} a^{6} - \frac{3}{7} a^{5} - \frac{1}{7} a^{4} - \frac{3}{7} a^{3} + \frac{3}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{7} a^{11} + \frac{1}{7} a^{8} - \frac{3}{7} a^{7} - \frac{2}{7} a^{6} - \frac{3}{7} a^{5} + \frac{1}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{12} - \frac{3}{7} a^{8} - \frac{1}{7} a^{7} + \frac{3}{7} a^{6} + \frac{3}{7} a^{5} - \frac{2}{7} a^{4} - \frac{3}{7} a^{3} - \frac{1}{7} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{13} - \frac{1}{7} a^{8} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} + \frac{2}{7} a^{4} + \frac{3}{7} a^{3} + \frac{1}{7} a^{2} + \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{2208935141} a^{14} + \frac{156905185}{2208935141} a^{13} + \frac{140199210}{2208935141} a^{12} + \frac{36539448}{2208935141} a^{11} - \frac{71402612}{2208935141} a^{10} - \frac{109948555}{2208935141} a^{9} + \frac{231484205}{2208935141} a^{8} + \frac{51920276}{315562163} a^{7} - \frac{203771023}{2208935141} a^{6} - \frac{198868886}{2208935141} a^{5} + \frac{1005114245}{2208935141} a^{4} + \frac{374781055}{2208935141} a^{3} + \frac{397619944}{2208935141} a^{2} - \frac{1047472032}{2208935141} a - \frac{538962771}{2208935141}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 678.185779099 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{3}\cdot(2\pi)^{6}\cdot 678.185779099 \cdot 1}{2\sqrt{334095024862954369}}\approx 0.288770534060$

Galois group

$C_3\times D_5$ (as 15T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 30
The 12 conjugacy class representatives for $D_5\times C_3$
Character table for $D_5\times C_3$

Intermediate fields

\(\Q(\zeta_{7})^+\), 5.1.14161.1

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

Sibling fields

Galois closure: Deg 30

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $15$ $15$ $15$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ $15$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
$17$17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$