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Properties

Label 15.11.1451077666...0000.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 281^{4}\cdot 44850007^{2}$
Root discriminant $754.13$
Ramified primes $2, 5, 7, 281, 44850007$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T92

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2248000, 15736000, 21018800, -45803000, -80928000, 18576960, 72393848, 21382724, -6284088, -1599876, 127584, 39784, -648, -360, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 360*x^13 - 648*x^12 + 39784*x^11 + 127584*x^10 - 1599876*x^9 - 6284088*x^8 + 21382724*x^7 + 72393848*x^6 + 18576960*x^5 - 80928000*x^4 - 45803000*x^3 + 21018800*x^2 + 15736000*x + 2248000)
 
gp: K = bnfinit(x^15 - 360*x^13 - 648*x^12 + 39784*x^11 + 127584*x^10 - 1599876*x^9 - 6284088*x^8 + 21382724*x^7 + 72393848*x^6 + 18576960*x^5 - 80928000*x^4 - 45803000*x^3 + 21018800*x^2 + 15736000*x + 2248000, 1)
 

Normalized defining polynomial

\( x^{15} - 360 x^{13} - 648 x^{12} + 39784 x^{11} + 127584 x^{10} - 1599876 x^{9} - 6284088 x^{8} + 21382724 x^{7} + 72393848 x^{6} + 18576960 x^{5} - 80928000 x^{4} - 45803000 x^{3} + 21018800 x^{2} + 15736000 x + 2248000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14510776668759878086782120121838473216000000=2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 281^{4}\cdot 44850007^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $754.13$
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:  $2, 5, 7, 281, 44850007$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{20} a^{11} + \frac{1}{10} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{10} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2}$, $\frac{1}{40} a^{12} + \frac{1}{20} a^{9} + \frac{1}{10} a^{8} + \frac{1}{10} a^{7} + \frac{1}{10} a^{6} - \frac{1}{5} a^{5} + \frac{1}{10} a^{4} + \frac{1}{5} a^{3}$, $\frac{1}{5200} a^{13} + \frac{3}{520} a^{12} + \frac{1}{65} a^{11} - \frac{87}{1300} a^{10} + \frac{43}{650} a^{9} + \frac{71}{1300} a^{8} + \frac{51}{1300} a^{7} + \frac{49}{650} a^{6} - \frac{289}{1300} a^{5} + \frac{31}{650} a^{4} + \frac{12}{65} a^{3} - \frac{31}{65} a^{2} - \frac{1}{2} a + \frac{6}{13}$, $\frac{1}{151077215114651938673129497572711508395728800} a^{14} + \frac{65520851521332996591051345745766578107}{944232594466574616707059359829446927473305} a^{13} + \frac{73305136114984417877518753606625193365847}{7553860755732596933656474878635575419786440} a^{12} + \frac{45862736291323091821115888929226173562957}{9442325944665746167070593598294469274733050} a^{11} + \frac{1022053035288549213597831239954745205868053}{18884651889331492334141187196588938549466100} a^{10} - \frac{158041103702152547911559387837035236991671}{9442325944665746167070593598294469274733050} a^{9} - \frac{2208709840023244275531543772277363563862069}{37769303778662984668282374393177877098932200} a^{8} - \frac{3843039260857291740417837691309391784722821}{18884651889331492334141187196588938549466100} a^{7} + \frac{1660479058869487772748008626623394068411501}{37769303778662984668282374393177877098932200} a^{6} - \frac{1259295753256114797421116936288669660108779}{18884651889331492334141187196588938549466100} a^{5} + \frac{83457154409415489687421245398910222289443}{1888465188933149233414118719658893854946610} a^{4} - \frac{454479985633543367117319406617003716833341}{944232594466574616707059359829446927473305} a^{3} + \frac{1040546047487567379425668304102985933312257}{3776930377866298466828237439317787709893220} a^{2} + \frac{150414337426326161505016732488374653753219}{377693037786629846682823743931778770989322} a + \frac{71194050656080600173073937759146080156971}{188846518893314923341411871965889385494661}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $12$
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:  \( 8314234047450000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T92:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 648000
The 55 conjugacy class representatives for [A(5)^3]3=A(5)wr3 are not computed
Character table for [A(5)^3]3=A(5)wr3 is not computed

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 18 sibling: data not computed
Degree 30 sibling: data not computed
Degree 36 sibling: data not computed
Degree 45 sibling: 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 $15$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.18.44$x^{12} - 6 x^{11} + 8 x^{10} + 6 x^{8} + 4 x^{6} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 8$$4$$3$$18$12T166$[2, 2, 2, 2, 2, 2]^{9}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed
281Data not computed
44850007Data not computed