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Properties

Label 15.15.2241536559...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{18}\cdot 3^{20}\cdot 5^{28}\cdot 37^{12}$
Root discriminant $3603.32$
Ramified primes $2, 3, 5, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_5^2 : C_3):C_4$ (as 15T17)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![354600210318775000, -2944424066062500, -38648421034218750, 5027100198340625, 417820767937500, -81279655885875, -1580943446250, 527107781250, 2441611500, -1741196875, -1319790, 3080250, 0, -2775, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2775*x^13 + 3080250*x^11 - 1319790*x^10 - 1741196875*x^9 + 2441611500*x^8 + 527107781250*x^7 - 1580943446250*x^6 - 81279655885875*x^5 + 417820767937500*x^4 + 5027100198340625*x^3 - 38648421034218750*x^2 - 2944424066062500*x + 354600210318775000)
 
gp: K = bnfinit(x^15 - 2775*x^13 + 3080250*x^11 - 1319790*x^10 - 1741196875*x^9 + 2441611500*x^8 + 527107781250*x^7 - 1580943446250*x^6 - 81279655885875*x^5 + 417820767937500*x^4 + 5027100198340625*x^3 - 38648421034218750*x^2 - 2944424066062500*x + 354600210318775000, 1)
 

Normalized defining polynomial

\( x^{15} - 2775 x^{13} + 3080250 x^{11} - 1319790 x^{10} - 1741196875 x^{9} + 2441611500 x^{8} + 527107781250 x^{7} - 1580943446250 x^{6} - 81279655885875 x^{5} + 417820767937500 x^{4} + 5027100198340625 x^{3} - 38648421034218750 x^{2} - 2944424066062500 x + 354600210318775000 \)

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:  \(224153655922799764834770035947265625000000000000000000=2^{18}\cdot 3^{20}\cdot 5^{28}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3603.32$
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:  $2, 3, 5, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{4} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{5920} a^{5} + \frac{3}{32} a^{3} - \frac{1}{4} a^{2} + \frac{5}{32} a + \frac{3}{16}$, $\frac{1}{5920} a^{6} + \frac{3}{32} a^{4} - \frac{3}{32} a^{2} + \frac{7}{16} a - \frac{1}{2}$, $\frac{1}{5920} a^{7} - \frac{1}{8} a^{3} + \frac{3}{16} a^{2} - \frac{7}{32} a - \frac{1}{16}$, $\frac{1}{118400} a^{8} - \frac{1}{23680} a^{7} - \frac{1}{23680} a^{5} - \frac{1}{32} a^{4} + \frac{51}{640} a^{3} + \frac{31}{128} a^{2} - \frac{1}{4} a + \frac{11}{32}$, $\frac{1}{118400} a^{9} - \frac{1}{23680} a^{7} - \frac{1}{23680} a^{6} - \frac{1}{23680} a^{5} - \frac{49}{640} a^{4} - \frac{3}{64} a^{3} - \frac{45}{128} a^{2} - \frac{1}{16} a - \frac{15}{32}$, $\frac{1}{443862656000} a^{10} - \frac{1}{239925760} a^{8} + \frac{7}{2593792} a^{6} - \frac{19781}{599814400} a^{5} - \frac{925}{1296896} a^{4} + \frac{19781}{648448} a^{3} + \frac{171125}{2593792} a^{2} + \frac{231203}{648448} a + \frac{113029}{648448}$, $\frac{1}{443862656000} a^{11} - \frac{1}{239925760} a^{9} + \frac{7}{2593792} a^{7} - \frac{19781}{599814400} a^{6} - \frac{9013}{239925760} a^{5} + \frac{19781}{648448} a^{4} - \frac{153099}{2593792} a^{3} - \frac{93021}{648448} a^{2} + \frac{194085}{648448} a - \frac{1}{4}$, $\frac{1}{2219313280000} a^{12} - \frac{13}{12968960} a^{8} + \frac{182859}{2999072000} a^{7} + \frac{271}{23992576} a^{6} - \frac{2393}{119962880} a^{5} + \frac{79229}{2593792} a^{4} + \frac{21537}{648448} a^{3} + \frac{254843}{6484480} a^{2} - \frac{106069}{324224} a + \frac{38445}{324224}$, $\frac{1}{11001384492047360000} a^{13} + \frac{24379}{297334716001280000} a^{12} - \frac{13}{59466943200256000} a^{11} + \frac{1389}{2973347160012800} a^{10} + \frac{13}{64288587243520} a^{9} + \frac{181872322741}{59466943200256000} a^{8} + \frac{133340387669}{1607214681088000} a^{7} - \frac{808805249}{10045091756800} a^{6} + \frac{1662317719}{64288587243520} a^{5} - \frac{15850942537}{347505876992} a^{4} - \frac{5597079186743}{64288587243520} a^{3} + \frac{161240069377}{868764692480} a^{2} - \frac{25719643377}{86876469248} a - \frac{12242658439}{43438234624}$, $\frac{1}{55006922460236800000} a^{14} - \frac{7}{148667358000640000} a^{12} - \frac{33957}{59466943200256000} a^{11} + \frac{77}{1607214681088000} a^{10} - \frac{405712265557}{148667358000640000} a^{9} - \frac{21}{868764692480} a^{8} - \frac{25653750399}{1607214681088000} a^{7} - \frac{11685638097}{321442936217600} a^{6} + \frac{7229514567}{160721468108800} a^{5} - \frac{1005095307413}{40180367027200} a^{4} - \frac{3350622481}{347505876992} a^{3} - \frac{335299110897}{868764692480} a^{2} - \frac{28994254427}{86876469248} a + \frac{19138909881}{43438234624}$

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:  \( 47821760164600000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:(C_3:C_4)$ (as 15T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 300
The 8 conjugacy class representatives for $(C_5^2 : C_3):C_4$
Character table for $(C_5^2 : C_3):C_4$

Intermediate fields

3.3.1620.1

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

Sibling fields

Degree 15 sibling: data not computed
Degree 25 sibling: data not computed
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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{5}$

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$$[\ ]_{3}^{2}$
2.12.16.18$x^{12} + x^{10} + 6 x^{8} - 3 x^{6} + 6 x^{4} + x^{2} - 3$$6$$2$$16$$C_3 : C_4$$[2]_{3}^{2}$
$3$3.3.4.4$x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2}$
3.12.16.30$x^{12} + 93 x^{11} + 351 x^{10} + 3 x^{9} + 126 x^{8} - 297 x^{7} + 171 x^{6} + 243 x^{5} - 324 x^{4} - 54 x^{3} + 162 x^{2} - 243 x + 324$$3$$4$$16$$C_3 : C_4$$[2]^{4}$
$5$5.5.9.5$x^{5} + 105$$5$$1$$9$$F_5$$[9/4]_{4}$
5.10.19.7$x^{10} - 20 x^{5} + 105$$10$$1$$19$$C_5^2 : C_4$$[7/4, 9/4]_{4}$
37Data not computed