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Properties

Label 15.15.4757687700...0625.2
Degree $15$
Signature $[15, 0]$
Discriminant $5^{24}\cdot 7^{10}\cdot 41^{4}$
Root discriminant $129.37$
Ramified primes $5, 7, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5\times C_5^2:C_3$ (as 15T25)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![189461, -36900, -1183260, 1004295, 947100, -994230, -261960, 350165, 28465, -54550, -894, 3825, -5, -110, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 110*x^13 - 5*x^12 + 3825*x^11 - 894*x^10 - 54550*x^9 + 28465*x^8 + 350165*x^7 - 261960*x^6 - 994230*x^5 + 947100*x^4 + 1004295*x^3 - 1183260*x^2 - 36900*x + 189461)
 
gp: K = bnfinit(x^15 - 110*x^13 - 5*x^12 + 3825*x^11 - 894*x^10 - 54550*x^9 + 28465*x^8 + 350165*x^7 - 261960*x^6 - 994230*x^5 + 947100*x^4 + 1004295*x^3 - 1183260*x^2 - 36900*x + 189461, 1)
 

Normalized defining polynomial

\( x^{15} - 110 x^{13} - 5 x^{12} + 3825 x^{11} - 894 x^{10} - 54550 x^{9} + 28465 x^{8} + 350165 x^{7} - 261960 x^{6} - 994230 x^{5} + 947100 x^{4} + 1004295 x^{3} - 1183260 x^{2} - 36900 x + 189461 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(47576877003281652927398681640625=5^{24}\cdot 7^{10}\cdot 41^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.37$
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:  $5, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{1137537236432197500181260734250094396427} a^{14} - \frac{136478090496906117208679213859044670534}{1137537236432197500181260734250094396427} a^{13} - \frac{127821408401362140474603233840893266056}{1137537236432197500181260734250094396427} a^{12} - \frac{291222876718568405534737951232202248596}{1137537236432197500181260734250094396427} a^{11} + \frac{330558612517473929008914981840652334175}{1137537236432197500181260734250094396427} a^{10} - \frac{2297380589586175097051893598352095136}{16021651217354894368750151186621047837} a^{9} + \frac{71610530039082722604409911573027634221}{1137537236432197500181260734250094396427} a^{8} + \frac{146389061143240616042929971884691489803}{1137537236432197500181260734250094396427} a^{7} + \frac{202504009107678184703900996413177758922}{1137537236432197500181260734250094396427} a^{6} - \frac{181949075562686115592006801419527295586}{1137537236432197500181260734250094396427} a^{5} + \frac{247487427951449871502334840266458120532}{1137537236432197500181260734250094396427} a^{4} + \frac{510087467382393914223705226831151875026}{1137537236432197500181260734250094396427} a^{3} + \frac{457632661623088157651338583009554351826}{1137537236432197500181260734250094396427} a^{2} + \frac{390682357088706905861698882106154904140}{1137537236432197500181260734250094396427} a - \frac{412128350021647551731127619779937618229}{1137537236432197500181260734250094396427}$

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:  $14$
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:  \( 78914062103.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5^2:C_3$ (as 15T25):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 375
The 55 conjugacy class representatives for $C_5\times C_5^2:C_3$ are not computed
Character table for $C_5\times C_5^2:C_3$ 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 15 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 $15$ $15$ R R $15$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}$ $15$ $15$ $15$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ $15$ $15$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ $15$ $15$ $15$

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
7Data not computed
$41$41.5.4.4$x^{5} + 8856$$5$$1$$4$$C_5$$[\ ]_{5}$
41.5.0.1$x^{5} - x + 7$$1$$5$$0$$C_5$$[\ ]^{5}$
41.5.0.1$x^{5} - x + 7$$1$$5$$0$$C_5$$[\ ]^{5}$