sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 57*x^10 + 226*x^9 + 906*x^8 - 3428*x^7 - 5159*x^6 + 18006*x^5 + 10833*x^4 - 34220*x^3 - 4958*x^2 + 20914*x - 3371)
gp: K = bnfinit(y^12 - 4*y^11 - 57*y^10 + 226*y^9 + 906*y^8 - 3428*y^7 - 5159*y^6 + 18006*y^5 + 10833*y^4 - 34220*y^3 - 4958*y^2 + 20914*y - 3371, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - 57*x^10 + 226*x^9 + 906*x^8 - 3428*x^7 - 5159*x^6 + 18006*x^5 + 10833*x^4 - 34220*x^3 - 4958*x^2 + 20914*x - 3371);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 - 57*x^10 + 226*x^9 + 906*x^8 - 3428*x^7 - 5159*x^6 + 18006*x^5 + 10833*x^4 - 34220*x^3 - 4958*x^2 + 20914*x - 3371)
x 12 − 4 x 11 − 57 x 10 + 226 x 9 + 906 x 8 − 3428 x 7 − 5159 x 6 + 18006 x 5 + ⋯ − 3371 x^{12} - 4 x^{11} - 57 x^{10} + 226 x^{9} + 906 x^{8} - 3428 x^{7} - 5159 x^{6} + 18006 x^{5} + \cdots - 3371 x 1 2 − 4 x 1 1 − 5 7 x 1 0 + 2 2 6 x 9 + 9 0 6 x 8 − 3 4 2 8 x 7 − 5 1 5 9 x 6 + 1 8 0 0 6 x 5 + ⋯ − 3 3 7 1
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Degree : 12 12 1 2
sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
Signature : [ 12 , 0 ] [12, 0] [ 1 2 , 0 ]
sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
Discriminant :
737698544105842578608128 737698544105842578608128 7 3 7 6 9 8 5 4 4 1 0 5 8 4 2 5 7 8 6 0 8 1 2 8 = 2 12 ⋅ 1 3 9 ⋅ 1 9 8 \medspace = 2^{12}\cdot 13^{9}\cdot 19^{8} = 2 1 2 ⋅ 1 3 9 ⋅ 1 9 8
sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant : 97.50 97.50 9 7 . 5 0
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant : 2 ⋅ 1 3 3 / 4 1 9 2 / 3 ≈ 97.49669871530843 2\cdot 13^{3/4}19^{2/3}\approx 97.49669871530843 2 ⋅ 1 3 3 / 4 1 9 2 / 3 ≈ 9 7 . 4 9 6 6 9 8 7 1 5 3 0 8 4 3 Ramified primes :
2 2 2 13 13 1 3 19 19 1 9
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
Discriminant root field : Q ( 13 ) \Q(\sqrt{13}) Q ( 1 3 )
Aut ( K / Q ) \Aut(K/\Q) A u t ( K / Q ) = = = Gal ( K / Q ) \Gal(K/\Q) G a l ( K / Q ) C 12 C_{12} C 1 2
sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
This field is Galois and abelian over Q \Q Q Conductor : 988 = 2 2 ⋅ 13 ⋅ 19 988=2^{2}\cdot 13\cdot 19 9 8 8 = 2 2 ⋅ 1 3 ⋅ 1 9 Dirichlet character group :
{ \lbrace { χ 988 ( 1 , ⋅ ) \chi_{988}(1,·) χ 9 8 8 ( 1 , ⋅ ) χ 988 ( 805 , ⋅ ) \chi_{988}(805,·) χ 9 8 8 ( 8 0 5 , ⋅ ) χ 988 ( 961 , ⋅ ) \chi_{988}(961,·) χ 9 8 8 ( 9 6 1 , ⋅ ) χ 988 ( 619 , ⋅ ) \chi_{988}(619,·) χ 9 8 8 ( 6 1 9 , ⋅ ) χ 988 ( 77 , ⋅ ) \chi_{988}(77,·) χ 9 8 8 ( 7 7 , ⋅ ) χ 988 ( 463 , ⋅ ) \chi_{988}(463,·) χ 9 8 8 ( 4 6 3 , ⋅ ) χ 988 ( 83 , ⋅ ) \chi_{988}(83,·) χ 9 8 8 ( 8 3 , ⋅ ) χ 988 ( 885 , ⋅ ) \chi_{988}(885,·) χ 9 8 8 ( 8 8 5 , ⋅ ) χ 988 ( 723 , ⋅ ) \chi_{988}(723,·) χ 9 8 8 ( 7 2 3 , ⋅ ) χ 988 ( 343 , ⋅ ) \chi_{988}(343,·) χ 9 8 8 ( 3 4 3 , ⋅ ) χ 988 ( 729 , ⋅ ) \chi_{988}(729,·) χ 9 8 8 ( 7 2 9 , ⋅ ) χ 988 ( 239 , ⋅ ) \chi_{988}(239,·) χ 9 8 8 ( 2 3 9 , ⋅ ) } \rbrace } This is not a CM field . This field has no CM subfields.
1 1 1 a a a a 2 a^{2} a 2 a 3 a^{3} a 3 a 4 a^{4} a 4 a 5 a^{5} a 5 1 3 a 6 + 1 3 a 5 + 1 3 a 4 + 1 3 a 3 + 1 3 a − 1 3 \frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{3} 3 1 a 6 + 3 1 a 5 + 3 1 a 4 + 3 1 a 3 + 3 1 a − 3 1 1 3 a 7 − 1 3 a 3 + 1 3 a 2 + 1 3 a + 1 3 \frac{1}{3}a^{7}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3} 3 1 a 7 − 3 1 a 3 + 3 1 a 2 + 3 1 a + 3 1 1 3 a 8 − 1 3 a 4 + 1 3 a 3 + 1 3 a 2 + 1 3 a \frac{1}{3}a^{8}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a 3 1 a 8 − 3 1 a 4 + 3 1 a 3 + 3 1 a 2 + 3 1 a 1 3 a 9 − 1 3 a 5 + 1 3 a 4 + 1 3 a 3 + 1 3 a 2 \frac{1}{3}a^{9}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2} 3 1 a 9 − 3 1 a 5 + 3 1 a 4 + 3 1 a 3 + 3 1 a 2 1 645501 a 10 + 73201 645501 a 9 + 32779 215167 a 8 + 20743 215167 a 7 − 5292 215167 a 6 − 255293 645501 a 5 − 57275 215167 a 4 − 69142 215167 a 3 + 307207 645501 a 2 − 180713 645501 a + 225803 645501 \frac{1}{645501}a^{10}+\frac{73201}{645501}a^{9}+\frac{32779}{215167}a^{8}+\frac{20743}{215167}a^{7}-\frac{5292}{215167}a^{6}-\frac{255293}{645501}a^{5}-\frac{57275}{215167}a^{4}-\frac{69142}{215167}a^{3}+\frac{307207}{645501}a^{2}-\frac{180713}{645501}a+\frac{225803}{645501} 6 4 5 5 0 1 1 a 1 0 + 6 4 5 5 0 1 7 3 2 0 1 a 9 + 2 1 5 1 6 7 3 2 7 7 9 a 8 + 2 1 5 1 6 7 2 0 7 4 3 a 7 − 2 1 5 1 6 7 5 2 9 2 a 6 − 6 4 5 5 0 1 2 5 5 2 9 3 a 5 − 2 1 5 1 6 7 5 7 2 7 5 a 4 − 2 1 5 1 6 7 6 9 1 4 2 a 3 + 6 4 5 5 0 1 3 0 7 2 0 7 a 2 − 6 4 5 5 0 1 1 8 0 7 1 3 a + 6 4 5 5 0 1 2 2 5 8 0 3 1 14337442826841 a 11 − 7656815 14337442826841 a 10 + 441404369723 4779147608947 a 9 − 1401821421569 14337442826841 a 8 − 96516588300 4779147608947 a 7 − 1657275285320 14337442826841 a 6 + 16745023499 4779147608947 a 5 + 3107172893549 14337442826841 a 4 − 3662257903795 14337442826841 a 3 + 5062562272355 14337442826841 a 2 + 1814702873909 4779147608947 a + 1504049512114 4779147608947 \frac{1}{14337442826841}a^{11}-\frac{7656815}{14337442826841}a^{10}+\frac{441404369723}{4779147608947}a^{9}-\frac{1401821421569}{14337442826841}a^{8}-\frac{96516588300}{4779147608947}a^{7}-\frac{1657275285320}{14337442826841}a^{6}+\frac{16745023499}{4779147608947}a^{5}+\frac{3107172893549}{14337442826841}a^{4}-\frac{3662257903795}{14337442826841}a^{3}+\frac{5062562272355}{14337442826841}a^{2}+\frac{1814702873909}{4779147608947}a+\frac{1504049512114}{4779147608947} 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 a 1 1 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 7 6 5 6 8 1 5 a 1 0 + 4 7 7 9 1 4 7 6 0 8 9 4 7 4 4 1 4 0 4 3 6 9 7 2 3 a 9 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 4 0 1 8 2 1 4 2 1 5 6 9 a 8 − 4 7 7 9 1 4 7 6 0 8 9 4 7 9 6 5 1 6 5 8 8 3 0 0 a 7 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 6 5 7 2 7 5 2 8 5 3 2 0 a 6 + 4 7 7 9 1 4 7 6 0 8 9 4 7 1 6 7 4 5 0 2 3 4 9 9 a 5 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 3 1 0 7 1 7 2 8 9 3 5 4 9 a 4 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 3 6 6 2 2 5 7 9 0 3 7 9 5 a 3 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 5 0 6 2 5 6 2 2 7 2 3 5 5 a 2 + 4 7 7 9 1 4 7 6 0 8 9 4 7 1 8 1 4 7 0 2 8 7 3 9 0 9 a + 4 7 7 9 1 4 7 6 0 8 9 4 7 1 5 0 4 0 4 9 5 1 2 1 1 4
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
Ideal class group : Trivial group, which has order 1 1 1 (assuming GRH )
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Narrow class group : C 2 C_{2} C 2 2 2 2 (assuming GRH )
sage: K.narrow_class_group().invariants()
gp: bnfnarrow(K)
magma: NarrowClassGroup(K);
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
Rank : 11 11 1 1
sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
Torsion generator :
− 1 -1 − 1 2 2 2
sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
Fundamental units :
2090364732 4779147608947 a 11 − 3064094550 4779147608947 a 10 − 139389370792 4779147608947 a 9 + 174503031618 4779147608947 a 8 + 3005023414536 4779147608947 a 7 − 2576592590670 4779147608947 a 6 − 26352852012630 4779147608947 a 5 + 13024930337517 4779147608947 a 4 + 89534830049210 4779147608947 a 3 − 23046622414719 4779147608947 a 2 − 82937068675536 4779147608947 a + 15961309882959 4779147608947 \frac{2090364732}{4779147608947}a^{11}-\frac{3064094550}{4779147608947}a^{10}-\frac{139389370792}{4779147608947}a^{9}+\frac{174503031618}{4779147608947}a^{8}+\frac{3005023414536}{4779147608947}a^{7}-\frac{2576592590670}{4779147608947}a^{6}-\frac{26352852012630}{4779147608947}a^{5}+\frac{13024930337517}{4779147608947}a^{4}+\frac{89534830049210}{4779147608947}a^{3}-\frac{23046622414719}{4779147608947}a^{2}-\frac{82937068675536}{4779147608947}a+\frac{15961309882959}{4779147608947} 4 7 7 9 1 4 7 6 0 8 9 4 7 2 0 9 0 3 6 4 7 3 2 a 1 1 − 4 7 7 9 1 4 7 6 0 8 9 4 7 3 0 6 4 0 9 4 5 5 0 a 1 0 − 4 7 7 9 1 4 7 6 0 8 9 4 7 1 3 9 3 8 9 3 7 0 7 9 2 a 9 + 4 7 7 9 1 4 7 6 0 8 9 4 7 1 7 4 5 0 3 0 3 1 6 1 8 a 8 + 4 7 7 9 1 4 7 6 0 8 9 4 7 3 0 0 5 0 2 3 4 1 4 5 3 6 a 7 − 4 7 7 9 1 4 7 6 0 8 9 4 7 2 5 7 6 5 9 2 5 9 0 6 7 0 a 6 − 4 7 7 9 1 4 7 6 0 8 9 4 7 2 6 3 5 2 8 5 2 0 1 2 6 3 0 a 5 + 4 7 7 9 1 4 7 6 0 8 9 4 7 1 3 0 2 4 9 3 0 3 3 7 5 1 7 a 4 + 4 7 7 9 1 4 7 6 0 8 9 4 7 8 9 5 3 4 8 3 0 0 4 9 2 1 0 a 3 − 4 7 7 9 1 4 7 6 0 8 9 4 7 2 3 0 4 6 6 2 2 4 1 4 7 1 9 a 2 − 4 7 7 9 1 4 7 6 0 8 9 4 7 8 2 9 3 7 0 6 8 6 7 5 5 3 6 a + 4 7 7 9 1 4 7 6 0 8 9 4 7 1 5 9 6 1 3 0 9 8 8 2 9 5 9 1221704656 4779147608947 a 11 − 3856674500 4779147608947 a 10 − 75654465484 4779147608947 a 9 + 224517601213 4779147608947 a 8 + 1436977514832 4779147608947 a 7 − 3621606614554 4779147608947 a 6 − 10973664396788 4779147608947 a 5 + 20913754151853 4779147608947 a 4 + 33622527537640 4779147608947 a 3 − 40820504142676 4779147608947 a 2 − 29673244707528 4779147608947 a + 23088434754151 4779147608947 \frac{1221704656}{4779147608947}a^{11}-\frac{3856674500}{4779147608947}a^{10}-\frac{75654465484}{4779147608947}a^{9}+\frac{224517601213}{4779147608947}a^{8}+\frac{1436977514832}{4779147608947}a^{7}-\frac{3621606614554}{4779147608947}a^{6}-\frac{10973664396788}{4779147608947}a^{5}+\frac{20913754151853}{4779147608947}a^{4}+\frac{33622527537640}{4779147608947}a^{3}-\frac{40820504142676}{4779147608947}a^{2}-\frac{29673244707528}{4779147608947}a+\frac{23088434754151}{4779147608947} 4 7 7 9 1 4 7 6 0 8 9 4 7 1 2 2 1 7 0 4 6 5 6 a 1 1 − 4 7 7 9 1 4 7 6 0 8 9 4 7 3 8 5 6 6 7 4 5 0 0 a 1 0 − 4 7 7 9 1 4 7 6 0 8 9 4 7 7 5 6 5 4 4 6 5 4 8 4 a 9 + 4 7 7 9 1 4 7 6 0 8 9 4 7 2 2 4 5 1 7 6 0 1 2 1 3 a 8 + 4 7 7 9 1 4 7 6 0 8 9 4 7 1 4 3 6 9 7 7 5 1 4 8 3 2 a 7 − 4 7 7 9 1 4 7 6 0 8 9 4 7 3 6 2 1 6 0 6 6 1 4 5 5 4 a 6 − 4 7 7 9 1 4 7 6 0 8 9 4 7 1 0 9 7 3 6 6 4 3 9 6 7 8 8 a 5 + 4 7 7 9 1 4 7 6 0 8 9 4 7 2 0 9 1 3 7 5 4 1 5 1 8 5 3 a 4 + 4 7 7 9 1 4 7 6 0 8 9 4 7 3 3 6 2 2 5 2 7 5 3 7 6 4 0 a 3 − 4 7 7 9 1 4 7 6 0 8 9 4 7 4 0 8 2 0 5 0 4 1 4 2 6 7 6 a 2 − 4 7 7 9 1 4 7 6 0 8 9 4 7 2 9 6 7 3 2 4 4 7 0 7 5 2 8 a + 4 7 7 9 1 4 7 6 0 8 9 4 7 2 3 0 8 8 4 3 4 7 5 4 1 5 1 532 645501 a 11 − 1996 645501 a 10 − 30232 645501 a 9 + 110161 645501 a 8 + 159308 215167 a 7 − 1564918 645501 a 6 − 2710418 645501 a 5 + 6964621 645501 a 4 + 5879602 645501 a 3 − 8587763 645501 a 2 − 3404932 645501 a + 1028494 645501 \frac{532}{645501}a^{11}-\frac{1996}{645501}a^{10}-\frac{30232}{645501}a^{9}+\frac{110161}{645501}a^{8}+\frac{159308}{215167}a^{7}-\frac{1564918}{645501}a^{6}-\frac{2710418}{645501}a^{5}+\frac{6964621}{645501}a^{4}+\frac{5879602}{645501}a^{3}-\frac{8587763}{645501}a^{2}-\frac{3404932}{645501}a+\frac{1028494}{645501} 6 4 5 5 0 1 5 3 2 a 1 1 − 6 4 5 5 0 1 1 9 9 6 a 1 0 − 6 4 5 5 0 1 3 0 2 3 2 a 9 + 6 4 5 5 0 1 1 1 0 1 6 1 a 8 + 2 1 5 1 6 7 1 5 9 3 0 8 a 7 − 6 4 5 5 0 1 1 5 6 4 9 1 8 a 6 − 6 4 5 5 0 1 2 7 1 0 4 1 8 a 5 + 6 4 5 5 0 1 6 9 6 4 6 2 1 a 4 + 6 4 5 5 0 1 5 8 7 9 6 0 2 a 3 − 6 4 5 5 0 1 8 5 8 7 7 6 3 a 2 − 6 4 5 5 0 1 3 4 0 4 9 3 2 a + 6 4 5 5 0 1 1 0 2 8 4 9 4 2028655600 4779147608947 a 11 − 31117474466 14337442826841 a 10 − 311809041049 14337442826841 a 9 + 569808939520 4779147608947 a 8 + 1208181123280 4779147608947 a 7 − 8040750199635 4779147608947 a 6 − 2066725297097 4779147608947 a 5 + 103116500347240 14337442826841 a 4 − 26414986826783 14337442826841 a 3 − 95679509054206 14337442826841 a 2 + 30999625381549 14337442826841 a − 4565112859756 14337442826841 \frac{2028655600}{4779147608947}a^{11}-\frac{31117474466}{14337442826841}a^{10}-\frac{311809041049}{14337442826841}a^{9}+\frac{569808939520}{4779147608947}a^{8}+\frac{1208181123280}{4779147608947}a^{7}-\frac{8040750199635}{4779147608947}a^{6}-\frac{2066725297097}{4779147608947}a^{5}+\frac{103116500347240}{14337442826841}a^{4}-\frac{26414986826783}{14337442826841}a^{3}-\frac{95679509054206}{14337442826841}a^{2}+\frac{30999625381549}{14337442826841}a-\frac{4565112859756}{14337442826841} 4 7 7 9 1 4 7 6 0 8 9 4 7 2 0 2 8 6 5 5 6 0 0 a 1 1 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 3 1 1 1 7 4 7 4 4 6 6 a 1 0 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 3 1 1 8 0 9 0 4 1 0 4 9 a 9 + 4 7 7 9 1 4 7 6 0 8 9 4 7 5 6 9 8 0 8 9 3 9 5 2 0 a 8 + 4 7 7 9 1 4 7 6 0 8 9 4 7 1 2 0 8 1 8 1 1 2 3 2 8 0 a 7 − 4 7 7 9 1 4 7 6 0 8 9 4 7 8 0 4 0 7 5 0 1 9 9 6 3 5 a 6 − 4 7 7 9 1 4 7 6 0 8 9 4 7 2 0 6 6 7 2 5 2 9 7 0 9 7 a 5 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 0 3 1 1 6 5 0 0 3 4 7 2 4 0 a 4 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 2 6 4 1 4 9 8 6 8 2 6 7 8 3 a 3 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 9 5 6 7 9 5 0 9 0 5 4 2 0 6 a 2 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 3 0 9 9 9 6 2 5 3 8 1 5 4 9 a − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 4 5 6 5 1 1 2 8 5 9 7 5 6 18087527608 14337442826841 a 11 − 53526120286 14337442826841 a 10 − 1089661373488 14337442826841 a 9 + 2970332630755 14337442826841 a 8 + 6543467726564 4779147608947 a 7 − 42488705107048 14337442826841 a 6 − 139260574488428 14337442826841 a 5 + 193768362979312 14337442826841 a 4 + 399198335113912 14337442826841 a 3 − 259885599664340 14337442826841 a 2 − 338776754587261 14337442826841 a + 85065603426172 14337442826841 \frac{18087527608}{14337442826841}a^{11}-\frac{53526120286}{14337442826841}a^{10}-\frac{1089661373488}{14337442826841}a^{9}+\frac{2970332630755}{14337442826841}a^{8}+\frac{6543467726564}{4779147608947}a^{7}-\frac{42488705107048}{14337442826841}a^{6}-\frac{139260574488428}{14337442826841}a^{5}+\frac{193768362979312}{14337442826841}a^{4}+\frac{399198335113912}{14337442826841}a^{3}-\frac{259885599664340}{14337442826841}a^{2}-\frac{338776754587261}{14337442826841}a+\frac{85065603426172}{14337442826841} 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 8 0 8 7 5 2 7 6 0 8 a 1 1 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 5 3 5 2 6 1 2 0 2 8 6 a 1 0 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 0 8 9 6 6 1 3 7 3 4 8 8 a 9 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 2 9 7 0 3 3 2 6 3 0 7 5 5 a 8 + 4 7 7 9 1 4 7 6 0 8 9 4 7 6 5 4 3 4 6 7 7 2 6 5 6 4 a 7 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 4 2 4 8 8 7 0 5 1 0 7 0 4 8 a 6 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 3 9 2 6 0 5 7 4 4 8 8 4 2 8 a 5 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 9 3 7 6 8 3 6 2 9 7 9 3 1 2 a 4 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 3 9 9 1 9 8 3 3 5 1 1 3 9 1 2 a 3 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 2 5 9 8 8 5 5 9 9 6 6 4 3 4 0 a 2 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 3 3 8 7 7 6 7 5 4 5 8 7 2 6 1 a + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 8 5 0 6 5 6 0 3 4 2 6 1 7 2 9210453184 14337442826841 a 11 − 46711576486 14337442826841 a 10 − 480288545188 14337442826841 a 9 + 2596867244686 14337442826841 a 8 + 1970398412324 4779147608947 a 7 − 37893969406690 14337442826841 a 6 − 14064455603012 14337442826841 a 5 + 178360043409769 14337442826841 a 4 − 37143062568428 14337442826841 a 3 − 244067377604054 14337442826841 a 2 + 84163366170212 14337442826841 a + 58563048390871 14337442826841 \frac{9210453184}{14337442826841}a^{11}-\frac{46711576486}{14337442826841}a^{10}-\frac{480288545188}{14337442826841}a^{9}+\frac{2596867244686}{14337442826841}a^{8}+\frac{1970398412324}{4779147608947}a^{7}-\frac{37893969406690}{14337442826841}a^{6}-\frac{14064455603012}{14337442826841}a^{5}+\frac{178360043409769}{14337442826841}a^{4}-\frac{37143062568428}{14337442826841}a^{3}-\frac{244067377604054}{14337442826841}a^{2}+\frac{84163366170212}{14337442826841}a+\frac{58563048390871}{14337442826841} 1 4 3 3 7 4 4 2 8 2 6 8 4 1 9 2 1 0 4 5 3 1 8 4 a 1 1 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 4 6 7 1 1 5 7 6 4 8 6 a 1 0 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 4 8 0 2 8 8 5 4 5 1 8 8 a 9 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 2 5 9 6 8 6 7 2 4 4 6 8 6 a 8 + 4 7 7 9 1 4 7 6 0 8 9 4 7 1 9 7 0 3 9 8 4 1 2 3 2 4 a 7 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 3 7 8 9 3 9 6 9 4 0 6 6 9 0 a 6 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 4 0 6 4 4 5 5 6 0 3 0 1 2 a 5 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 7 8 3 6 0 0 4 3 4 0 9 7 6 9 a 4 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 3 7 1 4 3 0 6 2 5 6 8 4 2 8 a 3 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 2 4 4 0 6 7 3 7 7 6 0 4 0 5 4 a 2 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 8 4 1 6 3 3 6 6 1 7 0 2 1 2 a + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 5 8 5 6 3 0 4 8 3 9 0 8 7 1 18087527608 14337442826841 a 11 − 53526120286 14337442826841 a 10 − 1089661373488 14337442826841 a 9 + 2970332630755 14337442826841 a 8 + 6543467726564 4779147608947 a 7 − 42488705107048 14337442826841 a 6 − 139260574488428 14337442826841 a 5 + 193768362979312 14337442826841 a 4 + 399198335113912 14337442826841 a 3 − 259885599664340 14337442826841 a 2 − 324439311760420 14337442826841 a + 113740489079854 14337442826841 \frac{18087527608}{14337442826841}a^{11}-\frac{53526120286}{14337442826841}a^{10}-\frac{1089661373488}{14337442826841}a^{9}+\frac{2970332630755}{14337442826841}a^{8}+\frac{6543467726564}{4779147608947}a^{7}-\frac{42488705107048}{14337442826841}a^{6}-\frac{139260574488428}{14337442826841}a^{5}+\frac{193768362979312}{14337442826841}a^{4}+\frac{399198335113912}{14337442826841}a^{3}-\frac{259885599664340}{14337442826841}a^{2}-\frac{324439311760420}{14337442826841}a+\frac{113740489079854}{14337442826841} 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 8 0 8 7 5 2 7 6 0 8 a 1 1 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 5 3 5 2 6 1 2 0 2 8 6 a 1 0 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 0 8 9 6 6 1 3 7 3 4 8 8 a 9 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 2 9 7 0 3 3 2 6 3 0 7 5 5 a 8 + 4 7 7 9 1 4 7 6 0 8 9 4 7 6 5 4 3 4 6 7 7 2 6 5 6 4 a 7 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 4 2 4 8 8 7 0 5 1 0 7 0 4 8 a 6 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 3 9 2 6 0 5 7 4 4 8 8 4 2 8 a 5 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 9 3 7 6 8 3 6 2 9 7 9 3 1 2 a 4 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 3 9 9 1 9 8 3 3 5 1 1 3 9 1 2 a 3 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 2 5 9 8 8 5 5 9 9 6 6 4 3 4 0 a 2 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 3 2 4 4 3 9 3 1 1 7 6 0 4 2 0 a + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 1 3 7 4 0 4 8 9 0 7 9 8 5 4 7097802133 14337442826841 a 11 − 10201058147 14337442826841 a 10 − 158084259148 4779147608947 a 9 + 495076061200 14337442826841 a 8 + 3552599124836 4779147608947 a 7 − 4943019662237 14337442826841 a 6 − 35900875527735 4779147608947 a 5 + 21654648030194 14337442826841 a 4 + 422875717422086 14337442826841 a 3 − 59201395876513 14337442826841 a 2 − 158633858256350 4779147608947 a + 36584402348744 4779147608947 \frac{7097802133}{14337442826841}a^{11}-\frac{10201058147}{14337442826841}a^{10}-\frac{158084259148}{4779147608947}a^{9}+\frac{495076061200}{14337442826841}a^{8}+\frac{3552599124836}{4779147608947}a^{7}-\frac{4943019662237}{14337442826841}a^{6}-\frac{35900875527735}{4779147608947}a^{5}+\frac{21654648030194}{14337442826841}a^{4}+\frac{422875717422086}{14337442826841}a^{3}-\frac{59201395876513}{14337442826841}a^{2}-\frac{158633858256350}{4779147608947}a+\frac{36584402348744}{4779147608947} 1 4 3 3 7 4 4 2 8 2 6 8 4 1 7 0 9 7 8 0 2 1 3 3 a 1 1 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 0 2 0 1 0 5 8 1 4 7 a 1 0 − 4 7 7 9 1 4 7 6 0 8 9 4 7 1 5 8 0 8 4 2 5 9 1 4 8 a 9 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 4 9 5 0 7 6 0 6 1 2 0 0 a 8 + 4 7 7 9 1 4 7 6 0 8 9 4 7 3 5 5 2 5 9 9 1 2 4 8 3 6 a 7 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 4 9 4 3 0 1 9 6 6 2 2 3 7 a 6 − 4 7 7 9 1 4 7 6 0 8 9 4 7 3 5 9 0 0 8 7 5 5 2 7 7 3 5 a 5 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 2 1 6 5 4 6 4 8 0 3 0 1 9 4 a 4 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 4 2 2 8 7 5 7 1 7 4 2 2 0 8 6 a 3 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 5 9 2 0 1 3 9 5 8 7 6 5 1 3 a 2 − 4 7 7 9 1 4 7 6 0 8 9 4 7 1 5 8 6 3 3 8 5 8 2 5 6 3 5 0 a + 4 7 7 9 1 4 7 6 0 8 9 4 7 3 6 5 8 4 4 0 2 3 4 8 7 4 4 1021031987 14337442826841 a 11 − 6489865024 14337442826841 a 10 − 18185748521 4779147608947 a 9 + 121089446721 4779147608947 a 8 + 225993326489 4779147608947 a 7 − 1767609005736 4779147608947 a 6 − 857537097362 14337442826841 a 5 + 21775998353059 14337442826841 a 4 − 3248294236569 4779147608947 a 3 − 19791168895477 14337442826841 a 2 + 12617499427742 14337442826841 a + 416621411276 14337442826841 \frac{1021031987}{14337442826841}a^{11}-\frac{6489865024}{14337442826841}a^{10}-\frac{18185748521}{4779147608947}a^{9}+\frac{121089446721}{4779147608947}a^{8}+\frac{225993326489}{4779147608947}a^{7}-\frac{1767609005736}{4779147608947}a^{6}-\frac{857537097362}{14337442826841}a^{5}+\frac{21775998353059}{14337442826841}a^{4}-\frac{3248294236569}{4779147608947}a^{3}-\frac{19791168895477}{14337442826841}a^{2}+\frac{12617499427742}{14337442826841}a+\frac{416621411276}{14337442826841} 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 0 2 1 0 3 1 9 8 7 a 1 1 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 6 4 8 9 8 6 5 0 2 4 a 1 0 − 4 7 7 9 1 4 7 6 0 8 9 4 7 1 8 1 8 5 7 4 8 5 2 1 a 9 + 4 7 7 9 1 4 7 6 0 8 9 4 7 1 2 1 0 8 9 4 4 6 7 2 1 a 8 + 4 7 7 9 1 4 7 6 0 8 9 4 7 2 2 5 9 9 3 3 2 6 4 8 9 a 7 − 4 7 7 9 1 4 7 6 0 8 9 4 7 1 7 6 7 6 0 9 0 0 5 7 3 6 a 6 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 8 5 7 5 3 7 0 9 7 3 6 2 a 5 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 2 1 7 7 5 9 9 8 3 5 3 0 5 9 a 4 − 4 7 7 9 1 4 7 6 0 8 9 4 7 3 2 4 8 2 9 4 2 3 6 5 6 9 a 3 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 9 7 9 1 1 6 8 8 9 5 4 7 7 a 2 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 2 6 1 7 4 9 9 4 2 7 7 4 2 a + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 4 1 6 6 2 1 4 1 1 2 7 6 58398190795 14337442826841 a 11 − 319699344019 14337442826841 a 10 − 2922932347045 14337442826841 a 9 + 17714523452468 14337442826841 a 8 + 30644580476576 14337442826841 a 7 − 256219182276833 14337442826841 a 6 + 4151931287823 4779147608947 a 5 + 11 ⋯ 52 14337442826841 a 4 − 739895445751354 14337442826841 a 3 − 13 ⋯ 28 14337442826841 a 2 + 13 ⋯ 61 14337442826841 a − 213354330040193 14337442826841 \frac{58398190795}{14337442826841}a^{11}-\frac{319699344019}{14337442826841}a^{10}-\frac{2922932347045}{14337442826841}a^{9}+\frac{17714523452468}{14337442826841}a^{8}+\frac{30644580476576}{14337442826841}a^{7}-\frac{256219182276833}{14337442826841}a^{6}+\frac{4151931287823}{4779147608947}a^{5}+\frac{11\cdots 52}{14337442826841}a^{4}-\frac{739895445751354}{14337442826841}a^{3}-\frac{13\cdots 28}{14337442826841}a^{2}+\frac{13\cdots 61}{14337442826841}a-\frac{213354330040193}{14337442826841} 1 4 3 3 7 4 4 2 8 2 6 8 4 1 5 8 3 9 8 1 9 0 7 9 5 a 1 1 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 3 1 9 6 9 9 3 4 4 0 1 9 a 1 0 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 2 9 2 2 9 3 2 3 4 7 0 4 5 a 9 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 7 7 1 4 5 2 3 4 5 2 4 6 8 a 8 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 3 0 6 4 4 5 8 0 4 7 6 5 7 6 a 7 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 2 5 6 2 1 9 1 8 2 2 7 6 8 3 3 a 6 + 4 7 7 9 1 4 7 6 0 8 9 4 7 4 1 5 1 9 3 1 2 8 7 8 2 3 a 5 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 1 ⋯ 5 2 a 4 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 7 3 9 8 9 5 4 4 5 7 5 1 3 5 4 a 3 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 3 ⋯ 2 8 a 2 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 3 ⋯ 6 1 a − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 2 1 3 3 5 4 3 3 0 0 4 0 1 9 3 25187009245 4779147608947 a 11 − 90043361190 4779147608947 a 10 − 4417447077490 14337442826841 a 9 + 5015923658927 4779147608947 a 8 + 74982802589299 14337442826841 a 7 − 218791579664561 14337442826841 a 6 − 496906935667483 14337442826841 a 5 + 341319550524704 4779147608947 a 4 + 13 ⋯ 98 14337442826841 a 3 − 436542169599973 4779147608947 a 2 − 11 ⋯ 26 14337442826841 a + 79647700595407 4779147608947 \frac{25187009245}{4779147608947}a^{11}-\frac{90043361190}{4779147608947}a^{10}-\frac{4417447077490}{14337442826841}a^{9}+\frac{5015923658927}{4779147608947}a^{8}+\frac{74982802589299}{14337442826841}a^{7}-\frac{218791579664561}{14337442826841}a^{6}-\frac{496906935667483}{14337442826841}a^{5}+\frac{341319550524704}{4779147608947}a^{4}+\frac{13\cdots 98}{14337442826841}a^{3}-\frac{436542169599973}{4779147608947}a^{2}-\frac{11\cdots 26}{14337442826841}a+\frac{79647700595407}{4779147608947} 4 7 7 9 1 4 7 6 0 8 9 4 7 2 5 1 8 7 0 0 9 2 4 5 a 1 1 − 4 7 7 9 1 4 7 6 0 8 9 4 7 9 0 0 4 3 3 6 1 1 9 0 a 1 0 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 4 4 1 7 4 4 7 0 7 7 4 9 0 a 9 + 4 7 7 9 1 4 7 6 0 8 9 4 7 5 0 1 5 9 2 3 6 5 8 9 2 7 a 8 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 7 4 9 8 2 8 0 2 5 8 9 2 9 9 a 7 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 2 1 8 7 9 1 5 7 9 6 6 4 5 6 1 a 6 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 4 9 6 9 0 6 9 3 5 6 6 7 4 8 3 a 5 + 4 7 7 9 1 4 7 6 0 8 9 4 7 3 4 1 3 1 9 5 5 0 5 2 4 7 0 4 a 4 + 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 3 ⋯ 9 8 a 3 − 4 7 7 9 1 4 7 6 0 8 9 4 7 4 3 6 5 4 2 1 6 9 5 9 9 9 7 3 a 2 − 1 4 3 3 7 4 4 2 8 2 6 8 4 1 1 1 ⋯ 2 6 a + 4 7 7 9 1 4 7 6 0 8 9 4 7 7 9 6 4 7 7 0 0 5 9 5 4 0 7
(assuming GRH )
sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
Regulator : 59483998.235 59483998.235 5 9 4 8 3 9 9 8 . 2 3 5 (assuming GRH )
sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
lim s → 1 ( s − 1 ) ζ K ( s ) = ( 2 r 1 ⋅ ( 2 π ) r 2 ⋅ R ⋅ h w ⋅ ∣ D ∣ ≈ ( 2 12 ⋅ ( 2 π ) 0 ⋅ 59483998.235 ⋅ 1 2 ⋅ 737698544105842578608128 ≈ ( 0.14183736065
\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 59483998.235 \cdot 1}{2\cdot\sqrt{737698544105842578608128}}\cr\approx \mathstrut & 0.14183736065
\end{aligned} s → 1 lim ( s − 1 ) ζ K ( s ) = ( ≈ ( ≈ ( w ⋅ ∣ D ∣ 2 r 1 ⋅ ( 2 π ) r 2 ⋅ R ⋅ h 2 ⋅ 7 3 7 6 9 8 5 4 4 1 0 5 8 4 2 5 7 8 6 0 8 1 2 8 2 1 2 ⋅ ( 2 π ) 0 ⋅ 5 9 4 8 3 9 9 8 . 2 3 5 ⋅ 1 0 . 1 4 1 8 3 7 3 6 0 6 5 (assuming GRH )
sage: # self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 57*x^10 + 226*x^9 + 906*x^8 - 3428*x^7 - 5159*x^6 + 18006*x^5 + 10833*x^4 - 34220*x^3 - 4958*x^2 + 20914*x - 3371)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
gp: \\ self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^12 - 4*x^11 - 57*x^10 + 226*x^9 + 906*x^8 - 3428*x^7 - 5159*x^6 + 18006*x^5 + 10833*x^4 - 34220*x^3 - 4958*x^2 + 20914*x - 3371, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
magma: /* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - 57*x^10 + 226*x^9 + 906*x^8 - 3428*x^7 - 5159*x^6 + 18006*x^5 + 10833*x^4 - 34220*x^3 - 4958*x^2 + 20914*x - 3371);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
oscar: # self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 4*x^11 - 57*x^10 + 226*x^9 + 906*x^8 - 3428*x^7 - 5159*x^6 + 18006*x^5 + 10833*x^4 - 34220*x^3 - 4958*x^2 + 20914*x - 3371);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
C 12 C_{12} C 1 2 12T1 ):
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
Fields in the database are given up to isomorphism. Isomorphic
intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
p p p 2 2 2 3 3 3 5 5 5 7 7 7 11 11 1 1 13 13 1 3 17 17 1 7 19 19 1 9 23 23 2 3 29 29 2 9 31 31 3 1 37 37 3 7 41 41 4 1 43 43 4 3 47 47 4 7 53 53 5 3 59 59 5 9
Cycle type
R
6 2 {\href{/padicField/3.6.0.1}{6} }^{2} 6 2 12 {\href{/padicField/5.12.0.1}{12} } 1 2 4 3 {\href{/padicField/7.4.0.1}{4} }^{3} 4 3 4 3 {\href{/padicField/11.4.0.1}{4} }^{3} 4 3 R
6 2 {\href{/padicField/17.6.0.1}{6} }^{2} 6 2 R
3 4 {\href{/padicField/23.3.0.1}{3} }^{4} 3 4 3 4 {\href{/padicField/29.3.0.1}{3} }^{4} 3 4 4 3 {\href{/padicField/31.4.0.1}{4} }^{3} 4 3 4 3 {\href{/padicField/37.4.0.1}{4} }^{3} 4 3 12 {\href{/padicField/41.12.0.1}{12} } 1 2 3 4 {\href{/padicField/43.3.0.1}{3} }^{4} 3 4 12 {\href{/padicField/47.12.0.1}{12} } 1 2 3 4 {\href{/padicField/53.3.0.1}{3} }^{4} 3 4 12 {\href{/padicField/59.12.0.1}{12} } 1 2
In the table, R denotes a ramified prime.
Cycle lengths which are repeated in a cycle type are indicated by
exponents.
sage: # to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: \\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
magma: // to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
oscar: # to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
p p p Label Polynomial
e e e f f f c c c Galois group Slope content
2 2 2
2.6.2.12a1.2 x 12 + 2 x 10 + 2 x 9 + x 8 + 4 x 7 + 5 x 6 + 2 x 5 + 6 x 4 + 8 x 3 + x 2 + 4 x + 5 x^{12} + 2 x^{10} + 2 x^{9} + x^{8} + 4 x^{7} + 5 x^{6} + 2 x^{5} + 6 x^{4} + 8 x^{3} + x^{2} + 4 x + 5 x 1 2 + 2 x 1 0 + 2 x 9 + x 8 + 4 x 7 + 5 x 6 + 2 x 5 + 6 x 4 + 8 x 3 + x 2 + 4 x + 5 2 2 2 6 6 6 12 12 1 2 C 12 C_{12} C 1 2 [ 2 ] 6 [2]^{6} [ 2 ] 6
13 13 1 3
13.3.4.9a1.3 x 12 + 8 x 10 + 44 x 9 + 24 x 8 + 264 x 7 + 758 x 6 + 528 x 5 + 2920 x 4 + 5676 x 3 + 2904 x 2 + 10648 x + 14654 x^{12} + 8 x^{10} + 44 x^{9} + 24 x^{8} + 264 x^{7} + 758 x^{6} + 528 x^{5} + 2920 x^{4} + 5676 x^{3} + 2904 x^{2} + 10648 x + 14654 x 1 2 + 8 x 1 0 + 4 4 x 9 + 2 4 x 8 + 2 6 4 x 7 + 7 5 8 x 6 + 5 2 8 x 5 + 2 9 2 0 x 4 + 5 6 7 6 x 3 + 2 9 0 4 x 2 + 1 0 6 4 8 x + 1 4 6 5 4 4 4 4 3 3 3 9 9 9 C 12 C_{12} C 1 2 [ ] 4 3 [\ ]_{4}^{3} [ ] 4 3
19 19 1 9
19.4.3.8a1.3 x 12 + 6 x 10 + 33 x 9 + 18 x 8 + 132 x 7 + 395 x 6 + 264 x 5 + 762 x 4 + 1595 x 3 + 750 x 2 + 132 x + 27 x^{12} + 6 x^{10} + 33 x^{9} + 18 x^{8} + 132 x^{7} + 395 x^{6} + 264 x^{5} + 762 x^{4} + 1595 x^{3} + 750 x^{2} + 132 x + 27 x 1 2 + 6 x 1 0 + 3 3 x 9 + 1 8 x 8 + 1 3 2 x 7 + 3 9 5 x 6 + 2 6 4 x 5 + 7 6 2 x 4 + 1 5 9 5 x 3 + 7 5 0 x 2 + 1 3 2 x + 2 7 3 3 3 4 4 4 8 8 8 C 12 C_{12} C 1 2 [ ] 3 4 [\ ]_{3}^{4} [ ] 3 4
(0) (0) (2) (3) (5) (7) (11) (13) (17) (19) (23) (29) (31) (37) (41) (43) (47) (53) (59)
