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Properties

Label 15.3.18077489127...0000.1
Degree $15$
Signature $[3, 6]$
Discriminant $2^{10}\cdot 3^{13}\cdot 5^{15}\cdot 11^{13}\cdot 101^{5}$
Root discriminant $765.26$
Ramified primes $2, 3, 5, 11, 101$
Class number $25$ (GRH)
Class group $[5, 5]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-624218022, 284948010, 805024170, -294061995, 20538450, 20814100, -2066285, -2743665, 937980, -27435, -33870, 3705, 675, -105, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 - 105*x^13 + 675*x^12 + 3705*x^11 - 33870*x^10 - 27435*x^9 + 937980*x^8 - 2743665*x^7 - 2066285*x^6 + 20814100*x^5 + 20538450*x^4 - 294061995*x^3 + 805024170*x^2 + 284948010*x - 624218022)
 
gp: K = bnfinit(x^15 - 5*x^14 - 105*x^13 + 675*x^12 + 3705*x^11 - 33870*x^10 - 27435*x^9 + 937980*x^8 - 2743665*x^7 - 2066285*x^6 + 20814100*x^5 + 20538450*x^4 - 294061995*x^3 + 805024170*x^2 + 284948010*x - 624218022, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} - 105 x^{13} + 675 x^{12} + 3705 x^{11} - 33870 x^{10} - 27435 x^{9} + 937980 x^{8} - 2743665 x^{7} - 2066285 x^{6} + 20814100 x^{5} + 20538450 x^{4} - 294061995 x^{3} + 805024170 x^{2} + 284948010 x - 624218022 \)

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:  \(18077489127555014674740676881656250000000000=2^{10}\cdot 3^{13}\cdot 5^{15}\cdot 11^{13}\cdot 101^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $765.26$
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:  $2, 3, 5, 11, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{4}$, $\frac{1}{9} a^{9} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} - \frac{1}{9} a^{4}$, $\frac{1}{99} a^{10} - \frac{5}{99} a^{8} - \frac{4}{33} a^{6} - \frac{1}{9} a^{5} - \frac{10}{99} a^{4} + \frac{1}{3} a^{3} - \frac{2}{33} a^{2} - \frac{5}{11}$, $\frac{1}{297} a^{11} + \frac{2}{99} a^{9} + \frac{1}{27} a^{8} + \frac{7}{99} a^{7} + \frac{1}{9} a^{6} + \frac{1}{297} a^{5} - \frac{1}{9} a^{4} + \frac{1}{11} a^{3} + \frac{1}{3} a^{2} + \frac{2}{11} a$, $\frac{1}{297} a^{12} + \frac{1}{27} a^{9} - \frac{5}{99} a^{8} - \frac{1}{9} a^{7} - \frac{26}{297} a^{6} - \frac{1}{9} a^{5} + \frac{7}{99} a^{4} + \frac{1}{3} a^{3} + \frac{10}{33} a^{2} - \frac{1}{11}$, $\frac{1}{891} a^{13} - \frac{1}{891} a^{12} + \frac{1}{891} a^{11} + \frac{2}{891} a^{10} - \frac{20}{891} a^{9} + \frac{38}{891} a^{8} + \frac{28}{891} a^{7} + \frac{35}{891} a^{6} - \frac{4}{81} a^{5} + \frac{5}{33} a^{4} - \frac{14}{33} a^{3} + \frac{2}{11} a^{2} + \frac{1}{33} a + \frac{5}{33}$, $\frac{1}{1048025468149033164653407007250615528754994061} a^{14} + \frac{34695145952866758894812596781518341063682}{116447274238781462739267445250068392083888229} a^{13} - \frac{29087023432800562720918959464945567350241}{116447274238781462739267445250068392083888229} a^{12} - \frac{30248352366900265550814383817535754697116}{38815758079593820913089148416689464027962743} a^{11} - \frac{35763786387258256320795021147210829333849}{38815758079593820913089148416689464027962743} a^{10} + \frac{100630256925858537719308461936956276529614}{12938586026531273637696382805563154675987581} a^{9} - \frac{17597511208003542308776331711235895074812671}{349341822716344388217802335750205176251664687} a^{8} + \frac{1044124901860889217843721460681469520652168}{10586115839889223885387949568188035643989839} a^{7} + \frac{6743670098022923918720251283668910427102746}{116447274238781462739267445250068392083888229} a^{6} + \frac{50904488895785880888350748403894423820161210}{1048025468149033164653407007250615528754994061} a^{5} + \frac{16233308561853905571787073260294486705993037}{116447274238781462739267445250068392083888229} a^{4} - \frac{13426827567297668143309475789634889920267992}{38815758079593820913089148416689464027962743} a^{3} + \frac{12550730422858353486272741524277360491748560}{38815758079593820913089148416689464027962743} a^{2} + \frac{4282535391865330602976905993001085746943134}{12938586026531273637696382805563154675987581} a - \frac{3993359482326728808218470085699118439500997}{38815758079593820913089148416689464027962743}$

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

Class group and class number

$C_{5}\times C_{5}$, which has order $25$ (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:  $8$
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:  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:  \( 950606739633431.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_5$ (as 15T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.3.13332.1, 5.1.3706003125.4

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$3$3.5.4.1$x^{5} - 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$5.5.5.2$x^{5} + 5 x + 5$$5$$1$$5$$F_5$$[5/4]_{4}$
5.10.10.7$x^{10} + 10 x^{8} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 12$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
$11$11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.9$x^{10} + 297$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$101$$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 2$$1$$1$$0$Trivial$[\ ]$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.1$x^{2} - 101$$2$$1$$1$$C_2$$[\ ]_{2}$