Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 11*x^20 - 121*x^18 - 1441*x^16 + 110*x^14 + 39314*x^12 + 149490*x^10 + 224906*x^8 + 145937*x^6 + 38907*x^4 + 3267*x^2 - 1)
gp: K = bnfinit(y^22 + 11*y^20 - 121*y^18 - 1441*y^16 + 110*y^14 + 39314*y^12 + 149490*y^10 + 224906*y^8 + 145937*y^6 + 38907*y^4 + 3267*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 11*x^20 - 121*x^18 - 1441*x^16 + 110*x^14 + 39314*x^12 + 149490*x^10 + 224906*x^8 + 145937*x^6 + 38907*x^4 + 3267*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 11*x^20 - 121*x^18 - 1441*x^16 + 110*x^14 + 39314*x^12 + 149490*x^10 + 224906*x^8 + 145937*x^6 + 38907*x^4 + 3267*x^2 - 1)
\( x^{22} + 11 x^{20} - 121 x^{18} - 1441 x^{16} + 110 x^{14} + 39314 x^{12} + 149490 x^{10} + 224906 x^{8} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[6, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1580153645858805893796455628954412194463744\)
\(\medspace = 2^{36}\cdot 7^{10}\cdot 11^{22}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(82.82\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(7\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{3}$, $\frac{1}{21\!\cdots\!96}a^{20}-\frac{24\!\cdots\!41}{10\!\cdots\!98}a^{18}-\frac{37\!\cdots\!73}{21\!\cdots\!96}a^{16}+\frac{72\!\cdots\!99}{52\!\cdots\!49}a^{14}+\frac{82\!\cdots\!28}{52\!\cdots\!49}a^{12}-\frac{18\!\cdots\!05}{10\!\cdots\!98}a^{10}-\frac{1}{2}a^{9}-\frac{13\!\cdots\!77}{52\!\cdots\!49}a^{8}+\frac{20\!\cdots\!51}{52\!\cdots\!49}a^{6}+\frac{58\!\cdots\!23}{21\!\cdots\!96}a^{4}-\frac{59\!\cdots\!21}{52\!\cdots\!49}a^{2}-\frac{1}{2}a+\frac{70\!\cdots\!01}{21\!\cdots\!96}$, $\frac{1}{21\!\cdots\!96}a^{21}-\frac{24\!\cdots\!41}{10\!\cdots\!98}a^{19}-\frac{37\!\cdots\!73}{21\!\cdots\!96}a^{17}+\frac{72\!\cdots\!99}{52\!\cdots\!49}a^{15}+\frac{82\!\cdots\!28}{52\!\cdots\!49}a^{13}-\frac{18\!\cdots\!05}{10\!\cdots\!98}a^{11}-\frac{1}{2}a^{10}-\frac{13\!\cdots\!77}{52\!\cdots\!49}a^{9}+\frac{20\!\cdots\!51}{52\!\cdots\!49}a^{7}+\frac{58\!\cdots\!23}{21\!\cdots\!96}a^{5}-\frac{59\!\cdots\!21}{52\!\cdots\!49}a^{3}-\frac{1}{2}a^{2}+\frac{70\!\cdots\!01}{21\!\cdots\!96}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $13$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{34\!\cdots\!59}{10\!\cdots\!98}a^{21}+\frac{37\!\cdots\!21}{10\!\cdots\!98}a^{19}-\frac{20\!\cdots\!10}{52\!\cdots\!49}a^{17}-\frac{49\!\cdots\!93}{10\!\cdots\!98}a^{15}+\frac{31\!\cdots\!68}{52\!\cdots\!49}a^{13}+\frac{13\!\cdots\!63}{10\!\cdots\!98}a^{11}+\frac{25\!\cdots\!27}{52\!\cdots\!49}a^{9}+\frac{75\!\cdots\!77}{10\!\cdots\!98}a^{7}+\frac{46\!\cdots\!33}{10\!\cdots\!98}a^{5}+\frac{56\!\cdots\!37}{52\!\cdots\!49}a^{3}+\frac{35\!\cdots\!67}{52\!\cdots\!49}a$, $\frac{33\!\cdots\!01}{52\!\cdots\!49}a^{21}+\frac{13\!\cdots\!63}{52\!\cdots\!49}a^{19}-\frac{66\!\cdots\!61}{52\!\cdots\!49}a^{17}-\frac{36\!\cdots\!59}{10\!\cdots\!98}a^{15}+\frac{67\!\cdots\!51}{10\!\cdots\!98}a^{13}+\frac{22\!\cdots\!69}{10\!\cdots\!98}a^{11}-\frac{87\!\cdots\!73}{10\!\cdots\!98}a^{9}-\frac{48\!\cdots\!87}{10\!\cdots\!98}a^{7}-\frac{69\!\cdots\!65}{10\!\cdots\!98}a^{5}-\frac{30\!\cdots\!05}{10\!\cdots\!98}a^{3}-\frac{27\!\cdots\!71}{10\!\cdots\!98}a$, $\frac{28\!\cdots\!17}{10\!\cdots\!98}a^{20}+\frac{30\!\cdots\!83}{10\!\cdots\!98}a^{18}-\frac{34\!\cdots\!23}{10\!\cdots\!98}a^{16}-\frac{20\!\cdots\!78}{52\!\cdots\!49}a^{14}+\frac{41\!\cdots\!23}{52\!\cdots\!49}a^{12}+\frac{55\!\cdots\!65}{52\!\cdots\!49}a^{10}+\frac{20\!\cdots\!59}{52\!\cdots\!49}a^{8}+\frac{29\!\cdots\!23}{52\!\cdots\!49}a^{6}+\frac{35\!\cdots\!95}{10\!\cdots\!98}a^{4}+\frac{73\!\cdots\!63}{10\!\cdots\!98}a^{2}-\frac{45\!\cdots\!41}{10\!\cdots\!98}$, $\frac{30\!\cdots\!79}{52\!\cdots\!49}a^{20}+\frac{65\!\cdots\!05}{10\!\cdots\!98}a^{18}-\frac{37\!\cdots\!02}{52\!\cdots\!49}a^{16}-\frac{85\!\cdots\!61}{10\!\cdots\!98}a^{14}+\frac{12\!\cdots\!10}{52\!\cdots\!49}a^{12}+\frac{23\!\cdots\!97}{10\!\cdots\!98}a^{10}+\frac{42\!\cdots\!47}{52\!\cdots\!49}a^{8}+\frac{11\!\cdots\!03}{10\!\cdots\!98}a^{6}+\frac{32\!\cdots\!68}{52\!\cdots\!49}a^{4}+\frac{53\!\cdots\!22}{52\!\cdots\!49}a^{2}-\frac{20\!\cdots\!23}{52\!\cdots\!49}$, $\frac{93\!\cdots\!32}{52\!\cdots\!49}a^{20}+\frac{19\!\cdots\!51}{10\!\cdots\!98}a^{18}-\frac{23\!\cdots\!79}{10\!\cdots\!98}a^{16}-\frac{12\!\cdots\!19}{52\!\cdots\!49}a^{14}+\frac{58\!\cdots\!75}{52\!\cdots\!49}a^{12}+\frac{36\!\cdots\!04}{52\!\cdots\!49}a^{10}+\frac{12\!\cdots\!68}{52\!\cdots\!49}a^{8}+\frac{16\!\cdots\!10}{52\!\cdots\!49}a^{6}+\frac{75\!\cdots\!94}{52\!\cdots\!49}a^{4}+\frac{16\!\cdots\!99}{10\!\cdots\!98}a^{2}-\frac{19\!\cdots\!13}{10\!\cdots\!98}$, $a$, $\frac{34\!\cdots\!77}{10\!\cdots\!98}a^{21}+\frac{37\!\cdots\!03}{10\!\cdots\!98}a^{19}-\frac{42\!\cdots\!03}{10\!\cdots\!98}a^{17}-\frac{49\!\cdots\!43}{10\!\cdots\!98}a^{15}+\frac{65\!\cdots\!54}{52\!\cdots\!49}a^{13}+\frac{13\!\cdots\!23}{10\!\cdots\!98}a^{11}+\frac{24\!\cdots\!75}{52\!\cdots\!49}a^{9}+\frac{66\!\cdots\!59}{10\!\cdots\!98}a^{7}+\frac{32\!\cdots\!73}{10\!\cdots\!98}a^{5}+\frac{22\!\cdots\!66}{52\!\cdots\!49}a^{3}+\frac{13\!\cdots\!99}{10\!\cdots\!98}a$, $\frac{64\!\cdots\!25}{10\!\cdots\!98}a^{20}+\frac{68\!\cdots\!57}{10\!\cdots\!98}a^{18}-\frac{79\!\cdots\!73}{10\!\cdots\!98}a^{16}-\frac{44\!\cdots\!22}{52\!\cdots\!49}a^{14}+\frac{17\!\cdots\!38}{52\!\cdots\!49}a^{12}+\frac{12\!\cdots\!64}{52\!\cdots\!49}a^{10}+\frac{43\!\cdots\!51}{52\!\cdots\!49}a^{8}+\frac{59\!\cdots\!80}{52\!\cdots\!49}a^{6}+\frac{64\!\cdots\!13}{10\!\cdots\!98}a^{4}+\frac{12\!\cdots\!85}{10\!\cdots\!98}a^{2}+\frac{70\!\cdots\!89}{10\!\cdots\!98}$, $\frac{36\!\cdots\!09}{52\!\cdots\!49}a^{20}+\frac{65\!\cdots\!92}{52\!\cdots\!49}a^{18}-\frac{14\!\cdots\!41}{10\!\cdots\!98}a^{16}-\frac{55\!\cdots\!68}{52\!\cdots\!49}a^{14}+\frac{75\!\cdots\!97}{10\!\cdots\!98}a^{12}+\frac{55\!\cdots\!66}{52\!\cdots\!49}a^{10}-\frac{11\!\cdots\!73}{10\!\cdots\!98}a^{8}-\frac{16\!\cdots\!67}{52\!\cdots\!49}a^{6}-\frac{26\!\cdots\!11}{10\!\cdots\!98}a^{4}-\frac{33\!\cdots\!39}{52\!\cdots\!49}a^{2}-\frac{16\!\cdots\!38}{52\!\cdots\!49}$, $\frac{68\!\cdots\!83}{10\!\cdots\!98}a^{20}+\frac{34\!\cdots\!14}{52\!\cdots\!49}a^{18}-\frac{44\!\cdots\!26}{52\!\cdots\!49}a^{16}-\frac{89\!\cdots\!59}{10\!\cdots\!98}a^{14}+\frac{40\!\cdots\!92}{52\!\cdots\!49}a^{12}+\frac{25\!\cdots\!35}{10\!\cdots\!98}a^{10}+\frac{40\!\cdots\!67}{52\!\cdots\!49}a^{8}+\frac{97\!\cdots\!07}{10\!\cdots\!98}a^{6}+\frac{45\!\cdots\!53}{10\!\cdots\!98}a^{4}+\frac{63\!\cdots\!91}{10\!\cdots\!98}a^{2}+\frac{10\!\cdots\!65}{52\!\cdots\!49}$, $\frac{20\!\cdots\!77}{21\!\cdots\!96}a^{21}-\frac{35\!\cdots\!85}{21\!\cdots\!96}a^{20}+\frac{55\!\cdots\!19}{52\!\cdots\!49}a^{19}-\frac{19\!\cdots\!85}{10\!\cdots\!98}a^{18}-\frac{24\!\cdots\!47}{21\!\cdots\!96}a^{17}+\frac{42\!\cdots\!09}{21\!\cdots\!96}a^{16}-\frac{72\!\cdots\!96}{52\!\cdots\!49}a^{15}+\frac{25\!\cdots\!65}{10\!\cdots\!98}a^{14}+\frac{55\!\cdots\!11}{52\!\cdots\!49}a^{13}-\frac{19\!\cdots\!29}{10\!\cdots\!98}a^{12}+\frac{39\!\cdots\!89}{10\!\cdots\!98}a^{11}-\frac{69\!\cdots\!89}{10\!\cdots\!98}a^{10}+\frac{15\!\cdots\!87}{10\!\cdots\!98}a^{9}-\frac{26\!\cdots\!83}{10\!\cdots\!98}a^{8}+\frac{11\!\cdots\!57}{52\!\cdots\!49}a^{7}-\frac{39\!\cdots\!67}{10\!\cdots\!98}a^{6}+\frac{29\!\cdots\!19}{21\!\cdots\!96}a^{5}-\frac{51\!\cdots\!09}{21\!\cdots\!96}a^{4}+\frac{39\!\cdots\!73}{10\!\cdots\!98}a^{3}-\frac{34\!\cdots\!52}{52\!\cdots\!49}a^{2}+\frac{66\!\cdots\!21}{21\!\cdots\!96}a-\frac{11\!\cdots\!51}{21\!\cdots\!96}$, $\frac{22\!\cdots\!41}{21\!\cdots\!96}a^{21}-\frac{39\!\cdots\!23}{21\!\cdots\!96}a^{20}+\frac{12\!\cdots\!35}{10\!\cdots\!98}a^{19}-\frac{10\!\cdots\!58}{52\!\cdots\!49}a^{18}-\frac{27\!\cdots\!95}{21\!\cdots\!96}a^{17}+\frac{48\!\cdots\!85}{21\!\cdots\!96}a^{16}-\frac{82\!\cdots\!61}{52\!\cdots\!49}a^{15}+\frac{14\!\cdots\!36}{52\!\cdots\!49}a^{14}+\frac{12\!\cdots\!35}{10\!\cdots\!98}a^{13}-\frac{21\!\cdots\!79}{10\!\cdots\!98}a^{12}+\frac{44\!\cdots\!07}{10\!\cdots\!98}a^{11}-\frac{39\!\cdots\!95}{52\!\cdots\!49}a^{10}+\frac{85\!\cdots\!29}{52\!\cdots\!49}a^{9}-\frac{29\!\cdots\!61}{10\!\cdots\!98}a^{8}+\frac{12\!\cdots\!63}{52\!\cdots\!49}a^{7}-\frac{22\!\cdots\!96}{52\!\cdots\!49}a^{6}+\frac{33\!\cdots\!01}{21\!\cdots\!96}a^{5}-\frac{58\!\cdots\!03}{21\!\cdots\!96}a^{4}+\frac{22\!\cdots\!19}{52\!\cdots\!49}a^{3}-\frac{38\!\cdots\!83}{52\!\cdots\!49}a^{2}+\frac{74\!\cdots\!87}{21\!\cdots\!96}a-\frac{13\!\cdots\!35}{21\!\cdots\!96}$, $\frac{16\!\cdots\!96}{52\!\cdots\!49}a^{21}+\frac{11\!\cdots\!97}{21\!\cdots\!96}a^{20}+\frac{17\!\cdots\!46}{52\!\cdots\!49}a^{19}+\frac{30\!\cdots\!21}{52\!\cdots\!49}a^{18}-\frac{39\!\cdots\!53}{10\!\cdots\!98}a^{17}-\frac{13\!\cdots\!93}{21\!\cdots\!96}a^{16}-\frac{46\!\cdots\!77}{10\!\cdots\!98}a^{15}-\frac{81\!\cdots\!87}{10\!\cdots\!98}a^{14}+\frac{35\!\cdots\!15}{10\!\cdots\!98}a^{13}+\frac{60\!\cdots\!13}{10\!\cdots\!98}a^{12}+\frac{12\!\cdots\!95}{10\!\cdots\!98}a^{11}+\frac{11\!\cdots\!83}{52\!\cdots\!49}a^{10}+\frac{24\!\cdots\!57}{52\!\cdots\!49}a^{9}+\frac{84\!\cdots\!05}{10\!\cdots\!98}a^{8}+\frac{72\!\cdots\!73}{10\!\cdots\!98}a^{7}+\frac{12\!\cdots\!17}{10\!\cdots\!98}a^{6}+\frac{47\!\cdots\!83}{10\!\cdots\!98}a^{5}+\frac{16\!\cdots\!53}{21\!\cdots\!96}a^{4}+\frac{12\!\cdots\!87}{10\!\cdots\!98}a^{3}+\frac{10\!\cdots\!36}{52\!\cdots\!49}a^{2}+\frac{10\!\cdots\!37}{10\!\cdots\!98}a+\frac{36\!\cdots\!79}{21\!\cdots\!96}$
| 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: | \( 15251388917300 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{8}\cdot 15251388917300 \cdot 1}{2\cdot\sqrt{1580153645858805893796455628954412194463744}}\cr\approx \mathstrut & 0.943080276918413 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 + 11*x^20 - 121*x^18 - 1441*x^16 + 110*x^14 + 39314*x^12 + 149490*x^10 + 224906*x^8 + 145937*x^6 + 38907*x^4 + 3267*x^2 - 1)
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))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^22 + 11*x^20 - 121*x^18 - 1441*x^16 + 110*x^14 + 39314*x^12 + 149490*x^10 + 224906*x^8 + 145937*x^6 + 38907*x^4 + 3267*x^2 - 1, 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)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^22 + 11*x^20 - 121*x^18 - 1441*x^16 + 110*x^14 + 39314*x^12 + 149490*x^10 + 224906*x^8 + 145937*x^6 + 38907*x^4 + 3267*x^2 - 1);
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)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 11*x^20 - 121*x^18 - 1441*x^16 + 110*x^14 + 39314*x^12 + 149490*x^10 + 224906*x^8 + 145937*x^6 + 38907*x^4 + 3267*x^2 - 1);
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))))
Galois group
$C_2^{10}.F_{11}$ (as 22T34):
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)
| A solvable group of order 112640 |
| The 44 conjugacy class representatives for $C_2^{10}.F_{11}$ |
| Character table for $C_2^{10}.F_{11}$ is not computed |
Intermediate fields
| 11.11.4910318845910094848.1 |
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]
Sibling fields
| Degree 22 sibling: | data not computed |
| Degree 44 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $20{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.10.0.1}{10} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | R | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.10.0.1}{10} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $22$ | $22$ | $1$ | $36$ | |||
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 7.10.5.1 | $x^{10} + 2401 x^{2} - 67228$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
| 7.10.5.1 | $x^{10} + 2401 x^{2} - 67228$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(11\)
| 11.11.11.6 | $x^{11} + 11 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ |
| 11.11.11.6 | $x^{11} + 11 x + 11$ | $11$ | $1$ | $11$ | $F_{11}$ | $[11/10]_{10}$ |