Properties

Label 15.3.334095024862954369.1
Degree $15$
Signature $[3, 6]$
Discriminant $7^{12}\cdot 17^{6}$
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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Show commands for: Magma / SageMath / Pari/GP

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]);
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)

Normalized defining polynomial

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

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:  \(334095024862954369=7^{12}\cdot 17^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $14.73$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $7, 17$
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 \(a\))

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

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:  \( \frac{34796749}{2208935141} a^{14} + \frac{63071795}{2208935141} a^{13} - \frac{86108704}{2208935141} a^{12} + \frac{84797679}{2208935141} a^{11} - \frac{392890051}{2208935141} a^{10} - \frac{905322246}{2208935141} a^{9} - \frac{1884288083}{2208935141} a^{8} - \frac{44267348}{315562163} a^{7} - \frac{625010744}{2208935141} a^{6} + \frac{2256343034}{2208935141} a^{5} - \frac{3841303037}{2208935141} a^{4} + \frac{4312154271}{2208935141} a^{3} + \frac{919713390}{2208935141} a^{2} - \frac{1255045132}{2208935141} a - \frac{3545238008}{2208935141} \),  \( \frac{48710311}{315562163} a^{14} - \frac{82013641}{315562163} a^{13} + \frac{128597389}{315562163} a^{12} - \frac{216522843}{315562163} a^{11} + \frac{212379}{45080309} a^{10} - \frac{954496707}{315562163} a^{9} + \frac{64843152}{45080309} a^{8} - \frac{1265184821}{315562163} a^{7} + \frac{2328694817}{315562163} a^{6} - \frac{3269922340}{315562163} a^{5} + \frac{3586463521}{315562163} a^{4} - \frac{256273689}{45080309} a^{3} + \frac{1767428062}{315562163} a^{2} - \frac{289166928}{45080309} a + \frac{246247030}{315562163} \),  \( \frac{5700061}{45080309} a^{14} - \frac{67185092}{315562163} a^{13} + \frac{103262728}{315562163} a^{12} - \frac{171327400}{315562163} a^{11} - \frac{7909361}{315562163} a^{10} - \frac{776415737}{315562163} a^{9} + \frac{384082323}{315562163} a^{8} - \frac{1016475415}{315562163} a^{7} + \frac{1843409354}{315562163} a^{6} - \frac{2589857714}{315562163} a^{5} + \frac{2866941625}{315562163} a^{4} - \frac{1426404346}{315562163} a^{3} + \frac{1385802217}{315562163} a^{2} - \frac{1609571834}{315562163} a + \frac{652818627}{315562163} \),  \( \frac{356491850}{2208935141} a^{14} - \frac{572280699}{2208935141} a^{13} + \frac{870527907}{2208935141} a^{12} - \frac{1584994221}{2208935141} a^{11} - \frac{47231703}{2208935141} a^{10} - \frac{7141812198}{2208935141} a^{9} + \frac{3244700397}{2208935141} a^{8} - \frac{171234215}{45080309} a^{7} + \frac{18412473428}{2208935141} a^{6} - \frac{21674293955}{2208935141} a^{5} + \frac{26483556556}{2208935141} a^{4} - \frac{14055339958}{2208935141} a^{3} + \frac{15968057022}{2208935141} a^{2} - \frac{18292832899}{2208935141} a + \frac{2176049088}{2208935141} \),  \( \frac{284244736}{2208935141} a^{14} - \frac{520788040}{2208935141} a^{13} + \frac{712468109}{2208935141} a^{12} - \frac{1263732863}{2208935141} a^{11} + \frac{22603834}{2208935141} a^{10} - \frac{5288860714}{2208935141} a^{9} + \frac{3818393823}{2208935141} a^{8} - \frac{800680957}{315562163} a^{7} + \frac{15009433804}{2208935141} a^{6} - \frac{18818810984}{2208935141} a^{5} + \frac{20926370715}{2208935141} a^{4} - \frac{9979420386}{2208935141} a^{3} + \frac{8807319569}{2208935141} a^{2} - \frac{12688719766}{2208935141} a + \frac{3201360453}{2208935141} \),  \( \frac{34796749}{2208935141} a^{14} + \frac{63071795}{2208935141} a^{13} - \frac{86108704}{2208935141} a^{12} + \frac{84797679}{2208935141} a^{11} - \frac{392890051}{2208935141} a^{10} - \frac{905322246}{2208935141} a^{9} - \frac{1884288083}{2208935141} a^{8} - \frac{44267348}{315562163} a^{7} - \frac{625010744}{2208935141} a^{6} + \frac{2256343034}{2208935141} a^{5} - \frac{3841303037}{2208935141} a^{4} + \frac{4312154271}{2208935141} a^{3} + \frac{919713390}{2208935141} a^{2} - \frac{1255045132}{2208935141} a - \frac{1336302867}{2208935141} \),  \( \frac{553916156}{2208935141} a^{14} - \frac{814591636}{2208935141} a^{13} + \frac{1215810328}{2208935141} a^{12} - \frac{2081999847}{2208935141} a^{11} - \frac{617313189}{2208935141} a^{10} - \frac{10705606487}{2208935141} a^{9} + \frac{3124584207}{2208935141} a^{8} - \frac{1788986939}{315562163} a^{7} + \frac{23393183706}{2208935141} a^{6} - \frac{30652065420}{2208935141} a^{5} + \frac{30774562174}{2208935141} a^{4} - \frac{12514592019}{2208935141} a^{3} + \frac{13596901860}{2208935141} a^{2} - \frac{21063011880}{2208935141} a + \frac{407879985}{2208935141} \),  \( \frac{199374699}{2208935141} a^{14} - \frac{288943861}{2208935141} a^{13} + \frac{453091887}{2208935141} a^{12} - \frac{772294482}{2208935141} a^{11} - \frac{146495005}{2208935141} a^{10} - \frac{3979024613}{2208935141} a^{9} + \frac{1115019548}{2208935141} a^{8} - \frac{717673700}{315562163} a^{7} + \frac{8378973730}{2208935141} a^{6} - \frac{12330600332}{2208935141} a^{5} + \frac{11564692263}{2208935141} a^{4} - \frac{7036534188}{2208935141} a^{3} + \frac{5830661647}{2208935141} a^{2} - \frac{8938603865}{2208935141} a + \frac{1173219093}{2208935141} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 678.185779099 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

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

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
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: 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 $15$ $15$ $15$ 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.

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
$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}$