sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 22*x^10 - 48*x^9 + 108*x^8 - 132*x^7 + 170*x^6 - 140*x^5 + 107*x^4 - 12*x^3 - 12*x^2 + 12*x + 9)
gp: K = bnfinit(y^12 - 4*y^11 + 22*y^10 - 48*y^9 + 108*y^8 - 132*y^7 + 170*y^6 - 140*y^5 + 107*y^4 - 12*y^3 - 12*y^2 + 12*y + 9, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 + 22*x^10 - 48*x^9 + 108*x^8 - 132*x^7 + 170*x^6 - 140*x^5 + 107*x^4 - 12*x^3 - 12*x^2 + 12*x + 9);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 + 22*x^10 - 48*x^9 + 108*x^8 - 132*x^7 + 170*x^6 - 140*x^5 + 107*x^4 - 12*x^3 - 12*x^2 + 12*x + 9)
x 12 − 4 x 11 + 22 x 10 − 48 x 9 + 108 x 8 − 132 x 7 + 170 x 6 − 140 x 5 + 107 x 4 + ⋯ + 9 x^{12} - 4 x^{11} + 22 x^{10} - 48 x^{9} + 108 x^{8} - 132 x^{7} + 170 x^{6} - 140 x^{5} + 107 x^{4} + \cdots + 9 x 1 2 − 4 x 1 1 + 2 2 x 1 0 − 4 8 x 9 + 1 0 8 x 8 − 1 3 2 x 7 + 1 7 0 x 6 − 1 4 0 x 5 + 1 0 7 x 4 + ⋯ + 9
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 : [ 0 , 6 ] [0, 6] [ 0 , 6 ]
sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
Discriminant :
12950250637492224 12950250637492224 1 2 9 5 0 2 5 0 6 3 7 4 9 2 2 2 4
= 2 24 ⋅ 3 8 ⋅ 7 6 \medspace = 2^{24}\cdot 3^{8}\cdot 7^{6} = 2 2 4 ⋅ 3 8 ⋅ 7 6
sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant : 22.01 22.01 2 2 . 0 1
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 2 3 4 / 3 7 1 / 2 ≈ 45.79000429827802 2^{2}3^{4/3}7^{1/2}\approx 45.79000429827802 2 2 3 4 / 3 7 1 / 2 ≈ 4 5 . 7 9 0 0 0 4 2 9 8 2 7 8 0 2
Ramified primes :
2 2 2 , 3 3 3 , 7 7 7
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
Discriminant root field : Q \Q Q
Aut ( K / Q ) \Aut(K/\Q) A u t ( K / Q ) :
C 6 C_6 C 6
sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
This field is not Galois over Q \Q Q .
This is not a CM field .
Maximal CM subfield : Q ( − 2 , 7 ) \Q(\sqrt{-2}, \sqrt{7}) Q ( − 2 , 7 )
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 , a 6 a^{6} a 6 , 1 3 a 7 − 1 3 a 6 + 1 3 a 4 − 1 3 a 3 \frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3} 3 1 a 7 − 3 1 a 6 + 3 1 a 4 − 3 1 a 3 , 1 3 a 8 − 1 3 a 6 + 1 3 a 5 − 1 3 a 3 \frac{1}{3}a^{8}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3} 3 1 a 8 − 3 1 a 6 + 3 1 a 5 − 3 1 a 3 , 1 3 a 9 − 1 3 a 3 \frac{1}{3}a^{9}-\frac{1}{3}a^{3} 3 1 a 9 − 3 1 a 3 , 1 27 a 10 − 1 27 a 9 − 4 27 a 8 − 1 27 a 7 − 1 27 a 6 − 13 27 a 5 + 10 27 a 4 − 1 3 a 3 + 1 9 a 2 − 1 9 a − 1 3 \frac{1}{27}a^{10}-\frac{1}{27}a^{9}-\frac{4}{27}a^{8}-\frac{1}{27}a^{7}-\frac{1}{27}a^{6}-\frac{13}{27}a^{5}+\frac{10}{27}a^{4}-\frac{1}{3}a^{3}+\frac{1}{9}a^{2}-\frac{1}{9}a-\frac{1}{3} 2 7 1 a 1 0 − 2 7 1 a 9 − 2 7 4 a 8 − 2 7 1 a 7 − 2 7 1 a 6 − 2 7 1 3 a 5 + 2 7 1 0 a 4 − 3 1 a 3 + 9 1 a 2 − 9 1 a − 3 1 , 1 2577231 a 11 + 8842 859077 a 10 − 135620 2577231 a 9 + 102187 2577231 a 8 + 60565 2577231 a 7 + 601687 2577231 a 6 − 395084 859077 a 5 + 6530 48627 a 4 + 175183 859077 a 3 − 95981 286359 a 2 + 357608 859077 a − 6595 286359 \frac{1}{2577231}a^{11}+\frac{8842}{859077}a^{10}-\frac{135620}{2577231}a^{9}+\frac{102187}{2577231}a^{8}+\frac{60565}{2577231}a^{7}+\frac{601687}{2577231}a^{6}-\frac{395084}{859077}a^{5}+\frac{6530}{48627}a^{4}+\frac{175183}{859077}a^{3}-\frac{95981}{286359}a^{2}+\frac{357608}{859077}a-\frac{6595}{286359} 2 5 7 7 2 3 1 1 a 1 1 + 8 5 9 0 7 7 8 8 4 2 a 1 0 − 2 5 7 7 2 3 1 1 3 5 6 2 0 a 9 + 2 5 7 7 2 3 1 1 0 2 1 8 7 a 8 + 2 5 7 7 2 3 1 6 0 5 6 5 a 7 + 2 5 7 7 2 3 1 6 0 1 6 8 7 a 6 − 8 5 9 0 7 7 3 9 5 0 8 4 a 5 + 4 8 6 2 7 6 5 3 0 a 4 + 8 5 9 0 7 7 1 7 5 1 8 3 a 3 − 2 8 6 3 5 9 9 5 9 8 1 a 2 + 8 5 9 0 7 7 3 5 7 6 0 8 a − 2 8 6 3 5 9 6 5 9 5
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
Ideal class group : C 4 C_{4} C 4 , which has order 4 4 4
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Narrow class group : C 4 C_{4} C 4 , which has order 4 4 4
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 : 5 5 5
sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
Torsion generator :
− 1 -1 − 1
(order 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 :
37892 859077 a 11 − 754318 2577231 a 10 + 3597577 2577231 a 9 − 11779595 2577231 a 8 + 24961642 2577231 a 7 − 45388631 2577231 a 6 + 54093136 2577231 a 5 − 1249376 48627 a 4 + 1954781 95453 a 3 − 11588239 859077 a 2 + 3264223 859077 a + 185548 286359 \frac{37892}{859077}a^{11}-\frac{754318}{2577231}a^{10}+\frac{3597577}{2577231}a^{9}-\frac{11779595}{2577231}a^{8}+\frac{24961642}{2577231}a^{7}-\frac{45388631}{2577231}a^{6}+\frac{54093136}{2577231}a^{5}-\frac{1249376}{48627}a^{4}+\frac{1954781}{95453}a^{3}-\frac{11588239}{859077}a^{2}+\frac{3264223}{859077}a+\frac{185548}{286359} 8 5 9 0 7 7 3 7 8 9 2 a 1 1 − 2 5 7 7 2 3 1 7 5 4 3 1 8 a 1 0 + 2 5 7 7 2 3 1 3 5 9 7 5 7 7 a 9 − 2 5 7 7 2 3 1 1 1 7 7 9 5 9 5 a 8 + 2 5 7 7 2 3 1 2 4 9 6 1 6 4 2 a 7 − 2 5 7 7 2 3 1 4 5 3 8 8 6 3 1 a 6 + 2 5 7 7 2 3 1 5 4 0 9 3 1 3 6 a 5 − 4 8 6 2 7 1 2 4 9 3 7 6 a 4 + 9 5 4 5 3 1 9 5 4 7 8 1 a 3 − 8 5 9 0 7 7 1 1 5 8 8 2 3 9 a 2 + 8 5 9 0 7 7 3 2 6 4 2 2 3 a + 2 8 6 3 5 9 1 8 5 5 4 8 , 218251 2577231 a 11 − 375689 2577231 a 10 + 1085486 859077 a 9 − 377681 859077 a 8 + 969622 286359 a 7 + 2511215 859077 a 6 + 10372775 2577231 a 5 + 271054 48627 a 4 + 1215766 859077 a 3 + 4984738 859077 a 2 + 3258124 859077 a + 256801 286359 \frac{218251}{2577231}a^{11}-\frac{375689}{2577231}a^{10}+\frac{1085486}{859077}a^{9}-\frac{377681}{859077}a^{8}+\frac{969622}{286359}a^{7}+\frac{2511215}{859077}a^{6}+\frac{10372775}{2577231}a^{5}+\frac{271054}{48627}a^{4}+\frac{1215766}{859077}a^{3}+\frac{4984738}{859077}a^{2}+\frac{3258124}{859077}a+\frac{256801}{286359} 2 5 7 7 2 3 1 2 1 8 2 5 1 a 1 1 − 2 5 7 7 2 3 1 3 7 5 6 8 9 a 1 0 + 8 5 9 0 7 7 1 0 8 5 4 8 6 a 9 − 8 5 9 0 7 7 3 7 7 6 8 1 a 8 + 2 8 6 3 5 9 9 6 9 6 2 2 a 7 + 8 5 9 0 7 7 2 5 1 1 2 1 5 a 6 + 2 5 7 7 2 3 1 1 0 3 7 2 7 7 5 a 5 + 4 8 6 2 7 2 7 1 0 5 4 a 4 + 8 5 9 0 7 7 1 2 1 5 7 6 6 a 3 + 8 5 9 0 7 7 4 9 8 4 7 3 8 a 2 + 8 5 9 0 7 7 3 2 5 8 1 2 4 a + 2 8 6 3 5 9 2 5 6 8 0 1 , 242558 2577231 a 11 − 321980 859077 a 10 + 4928627 2577231 a 9 − 10410541 2577231 a 8 + 18921905 2577231 a 7 − 21205006 2577231 a 6 + 1628735 286359 a 5 − 293681 48627 a 4 − 1632973 859077 a 3 + 1554550 286359 a 2 − 3187016 859077 a − 354995 286359 \frac{242558}{2577231}a^{11}-\frac{321980}{859077}a^{10}+\frac{4928627}{2577231}a^{9}-\frac{10410541}{2577231}a^{8}+\frac{18921905}{2577231}a^{7}-\frac{21205006}{2577231}a^{6}+\frac{1628735}{286359}a^{5}-\frac{293681}{48627}a^{4}-\frac{1632973}{859077}a^{3}+\frac{1554550}{286359}a^{2}-\frac{3187016}{859077}a-\frac{354995}{286359} 2 5 7 7 2 3 1 2 4 2 5 5 8 a 1 1 − 8 5 9 0 7 7 3 2 1 9 8 0 a 1 0 + 2 5 7 7 2 3 1 4 9 2 8 6 2 7 a 9 − 2 5 7 7 2 3 1 1 0 4 1 0 5 4 1 a 8 + 2 5 7 7 2 3 1 1 8 9 2 1 9 0 5 a 7 − 2 5 7 7 2 3 1 2 1 2 0 5 0 0 6 a 6 + 2 8 6 3 5 9 1 6 2 8 7 3 5 a 5 − 4 8 6 2 7 2 9 3 6 8 1 a 4 − 8 5 9 0 7 7 1 6 3 2 9 7 3 a 3 + 2 8 6 3 5 9 1 5 5 4 5 5 0 a 2 − 8 5 9 0 7 7 3 1 8 7 0 1 6 a − 2 8 6 3 5 9 3 5 4 9 9 5 , 22544 859077 a 11 − 223609 2577231 a 10 + 1449631 2577231 a 9 − 2549606 2577231 a 8 + 7402825 2577231 a 7 − 5859503 2577231 a 6 + 11729590 2577231 a 5 − 18668 48627 a 4 + 816754 286359 a 3 + 637349 859077 a 2 + 626740 859077 a + 18829 286359 \frac{22544}{859077}a^{11}-\frac{223609}{2577231}a^{10}+\frac{1449631}{2577231}a^{9}-\frac{2549606}{2577231}a^{8}+\frac{7402825}{2577231}a^{7}-\frac{5859503}{2577231}a^{6}+\frac{11729590}{2577231}a^{5}-\frac{18668}{48627}a^{4}+\frac{816754}{286359}a^{3}+\frac{637349}{859077}a^{2}+\frac{626740}{859077}a+\frac{18829}{286359} 8 5 9 0 7 7 2 2 5 4 4 a 1 1 − 2 5 7 7 2 3 1 2 2 3 6 0 9 a 1 0 + 2 5 7 7 2 3 1 1 4 4 9 6 3 1 a 9 − 2 5 7 7 2 3 1 2 5 4 9 6 0 6 a 8 + 2 5 7 7 2 3 1 7 4 0 2 8 2 5 a 7 − 2 5 7 7 2 3 1 5 8 5 9 5 0 3 a 6 + 2 5 7 7 2 3 1 1 1 7 2 9 5 9 0 a 5 − 4 8 6 2 7 1 8 6 6 8 a 4 + 2 8 6 3 5 9 8 1 6 7 5 4 a 3 + 8 5 9 0 7 7 6 3 7 3 4 9 a 2 + 8 5 9 0 7 7 6 2 6 7 4 0 a + 2 8 6 3 5 9 1 8 8 2 9 , 32965 286359 a 11 − 36932 95453 a 10 + 687032 286359 a 9 − 1278257 286359 a 8 + 3466585 286359 a 7 − 3766847 286359 a 6 + 2240491 95453 a 5 − 91324 5403 a 4 + 5825918 286359 a 3 − 576487 95453 a 2 + 579485 95453 a + 208730 95453 \frac{32965}{286359}a^{11}-\frac{36932}{95453}a^{10}+\frac{687032}{286359}a^{9}-\frac{1278257}{286359}a^{8}+\frac{3466585}{286359}a^{7}-\frac{3766847}{286359}a^{6}+\frac{2240491}{95453}a^{5}-\frac{91324}{5403}a^{4}+\frac{5825918}{286359}a^{3}-\frac{576487}{95453}a^{2}+\frac{579485}{95453}a+\frac{208730}{95453} 2 8 6 3 5 9 3 2 9 6 5 a 1 1 − 9 5 4 5 3 3 6 9 3 2 a 1 0 + 2 8 6 3 5 9 6 8 7 0 3 2 a 9 − 2 8 6 3 5 9 1 2 7 8 2 5 7 a 8 + 2 8 6 3 5 9 3 4 6 6 5 8 5 a 7 − 2 8 6 3 5 9 3 7 6 6 8 4 7 a 6 + 9 5 4 5 3 2 2 4 0 4 9 1 a 5 − 5 4 0 3 9 1 3 2 4 a 4 + 2 8 6 3 5 9 5 8 2 5 9 1 8 a 3 − 9 5 4 5 3 5 7 6 4 8 7 a 2 + 9 5 4 5 3 5 7 9 4 8 5 a + 9 5 4 5 3 2 0 8 7 3 0
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 : 3833.8726435419103 3833.8726435419103 3 8 3 3 . 8 7 2 6 4 3 5 4 1 9 1 0 3
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 0 ⋅ ( 2 π ) 6 ⋅ 3833.8726435419103 ⋅ 4 2 ⋅ 12950250637492224 ≈ ( 4.14579478575610
\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^{0}\cdot(2\pi)^{6}\cdot 3833.8726435419103 \cdot 4}{2\cdot\sqrt{12950250637492224}}\cr\approx \mathstrut & 4.14579478575610
\end{aligned} s → 1 lim ( s − 1 ) ζ K ( s ) = ( ≈ ( ≈ ( w ⋅ ∣ D ∣ 2 r 1 ⋅ ( 2 π ) r 2 ⋅ R ⋅ h 2 ⋅ 1 2 9 5 0 2 5 0 6 3 7 4 9 2 2 2 4 2 0 ⋅ ( 2 π ) 6 ⋅ 3 8 3 3 . 8 7 2 6 4 3 5 4 1 9 1 0 3 ⋅ 4 4 . 1 4 5 7 9 4 7 8 5 7 5 6 1 0
sage: # self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 + 22*x^10 - 48*x^9 + 108*x^8 - 132*x^7 + 170*x^6 - 140*x^5 + 107*x^4 - 12*x^3 - 12*x^2 + 12*x + 9)
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 + 22*x^10 - 48*x^9 + 108*x^8 - 132*x^7 + 170*x^6 - 140*x^5 + 107*x^4 - 12*x^3 - 12*x^2 + 12*x + 9, 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 + 22*x^10 - 48*x^9 + 108*x^8 - 132*x^7 + 170*x^6 - 140*x^5 + 107*x^4 - 12*x^3 - 12*x^2 + 12*x + 9);
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 + 22*x^10 - 48*x^9 + 108*x^8 - 132*x^7 + 170*x^6 - 140*x^5 + 107*x^4 - 12*x^3 - 12*x^2 + 12*x + 9);
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 6 × S 3 C_6\times S_3 C 6 × S 3 (as 12T18 ):
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
R
6 , 2 3 {\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{3} 6 , 2 3
R
6 2 {\href{/padicField/11.6.0.1}{6} }^{2} 6 2
6 2 {\href{/padicField/13.6.0.1}{6} }^{2} 6 2
2 6 {\href{/padicField/17.2.0.1}{2} }^{6} 2 6
3 4 {\href{/padicField/19.3.0.1}{3} }^{4} 3 4
6 , 2 3 {\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3} 6 , 2 3
6 2 {\href{/padicField/29.6.0.1}{6} }^{2} 6 2
6 2 {\href{/padicField/31.6.0.1}{6} }^{2} 6 2
2 6 {\href{/padicField/37.2.0.1}{2} }^{6} 2 6
6 2 {\href{/padicField/41.6.0.1}{6} }^{2} 6 2
6 2 {\href{/padicField/43.6.0.1}{6} }^{2} 6 2
6 2 {\href{/padicField/47.6.0.1}{6} }^{2} 6 2
2 6 {\href{/padicField/53.2.0.1}{2} }^{6} 2 6
3 4 {\href{/padicField/59.3.0.1}{3} }^{4} 3 4
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]
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.
(0) (0) (2) (3) (5) (7) (11) (13) (17) (19) (23) (29) (31) (37) (41) (43) (47) (53) (59)