Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 56*x^12 - 42*x^11 + 1043*x^10 + 1190*x^9 - 7987*x^8 - 11184*x^7 + 22169*x^6 + 37282*x^5 - 5292*x^4 - 22190*x^3 - 3164*x^2 + 3500*x + 881)
gp: K = bnfinit(y^14 - 56*y^12 - 42*y^11 + 1043*y^10 + 1190*y^9 - 7987*y^8 - 11184*y^7 + 22169*y^6 + 37282*y^5 - 5292*y^4 - 22190*y^3 - 3164*y^2 + 3500*y + 881, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 56*x^12 - 42*x^11 + 1043*x^10 + 1190*x^9 - 7987*x^8 - 11184*x^7 + 22169*x^6 + 37282*x^5 - 5292*x^4 - 22190*x^3 - 3164*x^2 + 3500*x + 881);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 56*x^12 - 42*x^11 + 1043*x^10 + 1190*x^9 - 7987*x^8 - 11184*x^7 + 22169*x^6 + 37282*x^5 - 5292*x^4 - 22190*x^3 - 3164*x^2 + 3500*x + 881)
\( x^{14} - 56 x^{12} - 42 x^{11} + 1043 x^{10} + 1190 x^{9} - 7987 x^{8} - 11184 x^{7} + 22169 x^{6} + \cdots + 881 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $14$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[14, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(401774962552217617093885952\)
\(\medspace = 2^{21}\cdot 7^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(79.48\) | 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: | $2^{3/2}7^{12/7}\approx 79.48487409873638$ | ||
| Ramified primes: |
\(2\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $14$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(392=2^{3}\cdot 7^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{392}(1,·)$, $\chi_{392}(197,·)$, $\chi_{392}(225,·)$, $\chi_{392}(169,·)$, $\chi_{392}(29,·)$, $\chi_{392}(141,·)$, $\chi_{392}(337,·)$, $\chi_{392}(365,·)$, $\chi_{392}(113,·)$, $\chi_{392}(85,·)$, $\chi_{392}(281,·)$, $\chi_{392}(57,·)$, $\chi_{392}(253,·)$, $\chi_{392}(309,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{31}a^{10}-\frac{14}{31}a^{9}+\frac{7}{31}a^{8}-\frac{13}{31}a^{7}-\frac{2}{31}a^{6}-\frac{11}{31}a^{5}-\frac{6}{31}a^{4}+\frac{10}{31}a^{3}+\frac{6}{31}a^{2}-\frac{3}{31}a-\frac{6}{31}$, $\frac{1}{31}a^{11}-\frac{3}{31}a^{9}-\frac{8}{31}a^{8}+\frac{2}{31}a^{7}-\frac{8}{31}a^{6}-\frac{5}{31}a^{5}-\frac{12}{31}a^{4}-\frac{9}{31}a^{3}-\frac{12}{31}a^{2}+\frac{14}{31}a+\frac{9}{31}$, $\frac{1}{589}a^{12}+\frac{3}{589}a^{11}+\frac{5}{589}a^{10}+\frac{243}{589}a^{9}+\frac{189}{589}a^{8}-\frac{199}{589}a^{7}-\frac{200}{589}a^{6}+\frac{288}{589}a^{5}+\frac{3}{19}a^{4}-\frac{207}{589}a^{3}-\frac{222}{589}a^{2}-\frac{66}{589}a-\frac{238}{589}$, $\frac{1}{10\!\cdots\!19}a^{13}+\frac{16726229385387}{10\!\cdots\!19}a^{12}-\frac{714636006765791}{10\!\cdots\!19}a^{11}+\frac{3658673466200}{10\!\cdots\!19}a^{10}-\frac{17\!\cdots\!76}{10\!\cdots\!19}a^{9}+\frac{37\!\cdots\!15}{10\!\cdots\!19}a^{8}+\frac{53\!\cdots\!29}{10\!\cdots\!19}a^{7}-\frac{856164377068916}{10\!\cdots\!19}a^{6}+\frac{51\!\cdots\!66}{10\!\cdots\!19}a^{5}+\frac{33\!\cdots\!12}{10\!\cdots\!19}a^{4}-\frac{16\!\cdots\!59}{10\!\cdots\!19}a^{3}-\frac{45\!\cdots\!84}{10\!\cdots\!19}a^{2}-\frac{17\!\cdots\!83}{10\!\cdots\!19}a-\frac{24\!\cdots\!75}{10\!\cdots\!19}$
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: |
$\frac{27099323122522}{56\!\cdots\!01}a^{13}-\frac{6240804314542}{56\!\cdots\!01}a^{12}-\frac{15\!\cdots\!04}{56\!\cdots\!01}a^{11}-\frac{789343035600707}{56\!\cdots\!01}a^{10}+\frac{28\!\cdots\!00}{56\!\cdots\!01}a^{9}+\frac{25\!\cdots\!12}{56\!\cdots\!01}a^{8}-\frac{21\!\cdots\!22}{56\!\cdots\!01}a^{7}-\frac{24\!\cdots\!80}{56\!\cdots\!01}a^{6}+\frac{63\!\cdots\!02}{56\!\cdots\!01}a^{5}+\frac{82\!\cdots\!84}{56\!\cdots\!01}a^{4}-\frac{28\!\cdots\!52}{56\!\cdots\!01}a^{3}-\frac{43\!\cdots\!38}{56\!\cdots\!01}a^{2}+\frac{39\!\cdots\!01}{56\!\cdots\!01}a+\frac{18\!\cdots\!54}{183536902704271}$, $\frac{715623237597702}{10\!\cdots\!19}a^{13}-\frac{590697398281347}{10\!\cdots\!19}a^{12}-\frac{39\!\cdots\!12}{10\!\cdots\!19}a^{11}+\frac{25\!\cdots\!15}{10\!\cdots\!19}a^{10}+\frac{24\!\cdots\!74}{34\!\cdots\!49}a^{9}+\frac{24\!\cdots\!86}{10\!\cdots\!19}a^{8}-\frac{59\!\cdots\!64}{10\!\cdots\!19}a^{7}-\frac{31\!\cdots\!77}{10\!\cdots\!19}a^{6}+\frac{18\!\cdots\!96}{10\!\cdots\!19}a^{5}+\frac{11\!\cdots\!14}{10\!\cdots\!19}a^{4}-\frac{14\!\cdots\!28}{10\!\cdots\!19}a^{3}-\frac{62\!\cdots\!03}{10\!\cdots\!19}a^{2}+\frac{25\!\cdots\!38}{10\!\cdots\!19}a+\frac{98\!\cdots\!07}{10\!\cdots\!19}$, $\frac{36840777864506}{34\!\cdots\!49}a^{13}-\frac{342180360331675}{10\!\cdots\!19}a^{12}-\frac{63\!\cdots\!76}{10\!\cdots\!19}a^{11}-\frac{28\!\cdots\!27}{10\!\cdots\!19}a^{10}+\frac{11\!\cdots\!62}{10\!\cdots\!19}a^{9}+\frac{10\!\cdots\!16}{10\!\cdots\!19}a^{8}-\frac{30\!\cdots\!52}{34\!\cdots\!49}a^{7}-\frac{99\!\cdots\!67}{10\!\cdots\!19}a^{6}+\frac{91\!\cdots\!00}{34\!\cdots\!49}a^{5}+\frac{34\!\cdots\!01}{10\!\cdots\!19}a^{4}-\frac{16\!\cdots\!40}{10\!\cdots\!19}a^{3}-\frac{20\!\cdots\!67}{10\!\cdots\!19}a^{2}+\frac{25\!\cdots\!54}{10\!\cdots\!19}a+\frac{33\!\cdots\!79}{10\!\cdots\!19}$, $\frac{92510555117652}{10\!\cdots\!19}a^{13}+\frac{62468905911131}{10\!\cdots\!19}a^{12}-\frac{52\!\cdots\!04}{10\!\cdots\!19}a^{11}-\frac{73\!\cdots\!10}{10\!\cdots\!19}a^{10}+\frac{31\!\cdots\!38}{34\!\cdots\!49}a^{9}+\frac{17\!\cdots\!61}{10\!\cdots\!19}a^{8}-\frac{74\!\cdots\!50}{10\!\cdots\!19}a^{7}-\frac{15\!\cdots\!89}{10\!\cdots\!19}a^{6}+\frac{19\!\cdots\!02}{10\!\cdots\!19}a^{5}+\frac{52\!\cdots\!52}{10\!\cdots\!19}a^{4}-\frac{10\!\cdots\!40}{10\!\cdots\!19}a^{3}-\frac{11\!\cdots\!81}{34\!\cdots\!49}a^{2}-\frac{13\!\cdots\!08}{10\!\cdots\!19}a+\frac{52\!\cdots\!94}{10\!\cdots\!19}$, $\frac{19779280026198}{10\!\cdots\!19}a^{13}-\frac{167498702290210}{10\!\cdots\!19}a^{12}-\frac{10\!\cdots\!20}{10\!\cdots\!19}a^{11}+\frac{84\!\cdots\!49}{10\!\cdots\!19}a^{10}+\frac{26\!\cdots\!84}{10\!\cdots\!19}a^{9}-\frac{14\!\cdots\!01}{10\!\cdots\!19}a^{8}-\frac{33\!\cdots\!26}{10\!\cdots\!19}a^{7}+\frac{10\!\cdots\!62}{10\!\cdots\!19}a^{6}+\frac{20\!\cdots\!22}{10\!\cdots\!19}a^{5}-\frac{25\!\cdots\!43}{10\!\cdots\!19}a^{4}-\frac{51\!\cdots\!24}{10\!\cdots\!19}a^{3}-\frac{62\!\cdots\!12}{10\!\cdots\!19}a^{2}+\frac{89\!\cdots\!86}{10\!\cdots\!19}a+\frac{18\!\cdots\!97}{10\!\cdots\!19}$, $\frac{659527148162862}{10\!\cdots\!19}a^{13}-\frac{351999330723893}{10\!\cdots\!19}a^{12}-\frac{36\!\cdots\!18}{10\!\cdots\!19}a^{11}-\frac{81\!\cdots\!75}{10\!\cdots\!19}a^{10}+\frac{69\!\cdots\!06}{10\!\cdots\!19}a^{9}+\frac{42\!\cdots\!71}{10\!\cdots\!19}a^{8}-\frac{54\!\cdots\!52}{10\!\cdots\!19}a^{7}-\frac{45\!\cdots\!23}{10\!\cdots\!19}a^{6}+\frac{16\!\cdots\!96}{10\!\cdots\!19}a^{5}+\frac{16\!\cdots\!08}{10\!\cdots\!19}a^{4}-\frac{11\!\cdots\!44}{10\!\cdots\!19}a^{3}-\frac{95\!\cdots\!72}{10\!\cdots\!19}a^{2}+\frac{25\!\cdots\!72}{10\!\cdots\!19}a+\frac{13\!\cdots\!09}{10\!\cdots\!19}$, $\frac{530699783636568}{10\!\cdots\!19}a^{13}-\frac{343268871367780}{10\!\cdots\!19}a^{12}-\frac{29\!\cdots\!40}{10\!\cdots\!19}a^{11}-\frac{31\!\cdots\!06}{10\!\cdots\!19}a^{10}+\frac{56\!\cdots\!24}{10\!\cdots\!19}a^{9}+\frac{27\!\cdots\!94}{10\!\cdots\!19}a^{8}-\frac{46\!\cdots\!16}{10\!\cdots\!19}a^{7}-\frac{32\!\cdots\!18}{10\!\cdots\!19}a^{6}+\frac{15\!\cdots\!16}{10\!\cdots\!19}a^{5}+\frac{12\!\cdots\!07}{10\!\cdots\!19}a^{4}-\frac{14\!\cdots\!44}{10\!\cdots\!19}a^{3}-\frac{97\!\cdots\!42}{10\!\cdots\!19}a^{2}+\frac{33\!\cdots\!00}{10\!\cdots\!19}a+\frac{15\!\cdots\!52}{10\!\cdots\!19}$, $\frac{406142686667293}{10\!\cdots\!19}a^{13}-\frac{242452396847306}{10\!\cdots\!19}a^{12}-\frac{22\!\cdots\!39}{10\!\cdots\!19}a^{11}-\frac{37\!\cdots\!96}{10\!\cdots\!19}a^{10}+\frac{42\!\cdots\!54}{10\!\cdots\!19}a^{9}+\frac{12\!\cdots\!67}{56\!\cdots\!01}a^{8}-\frac{33\!\cdots\!17}{10\!\cdots\!19}a^{7}-\frac{26\!\cdots\!39}{10\!\cdots\!19}a^{6}+\frac{10\!\cdots\!83}{10\!\cdots\!19}a^{5}+\frac{92\!\cdots\!94}{10\!\cdots\!19}a^{4}-\frac{63\!\cdots\!03}{10\!\cdots\!19}a^{3}-\frac{42\!\cdots\!31}{10\!\cdots\!19}a^{2}+\frac{10\!\cdots\!98}{10\!\cdots\!19}a+\frac{49\!\cdots\!60}{10\!\cdots\!19}$, $\frac{37406395531743}{10\!\cdots\!19}a^{13}-\frac{12509462061755}{56\!\cdots\!01}a^{12}-\frac{21\!\cdots\!66}{10\!\cdots\!19}a^{11}+\frac{615601159419495}{56\!\cdots\!01}a^{10}+\frac{52\!\cdots\!89}{10\!\cdots\!19}a^{9}-\frac{19\!\cdots\!12}{10\!\cdots\!19}a^{8}-\frac{63\!\cdots\!54}{10\!\cdots\!19}a^{7}+\frac{13\!\cdots\!48}{10\!\cdots\!19}a^{6}+\frac{37\!\cdots\!77}{10\!\cdots\!19}a^{5}-\frac{30\!\cdots\!19}{10\!\cdots\!19}a^{4}-\frac{48\!\cdots\!08}{56\!\cdots\!01}a^{3}-\frac{15\!\cdots\!26}{10\!\cdots\!19}a^{2}+\frac{33\!\cdots\!00}{10\!\cdots\!19}a+\frac{10\!\cdots\!20}{10\!\cdots\!19}$, $\frac{167000366042371}{10\!\cdots\!19}a^{13}-\frac{191913916497128}{10\!\cdots\!19}a^{12}-\frac{93\!\cdots\!66}{10\!\cdots\!19}a^{11}+\frac{37\!\cdots\!09}{10\!\cdots\!19}a^{10}+\frac{18\!\cdots\!37}{10\!\cdots\!19}a^{9}-\frac{467186843887932}{10\!\cdots\!19}a^{8}-\frac{15\!\cdots\!07}{10\!\cdots\!19}a^{7}-\frac{36\!\cdots\!59}{10\!\cdots\!19}a^{6}+\frac{30\!\cdots\!56}{56\!\cdots\!01}a^{5}+\frac{22\!\cdots\!36}{10\!\cdots\!19}a^{4}-\frac{75\!\cdots\!89}{10\!\cdots\!19}a^{3}-\frac{34\!\cdots\!95}{10\!\cdots\!19}a^{2}+\frac{22\!\cdots\!63}{10\!\cdots\!19}a+\frac{10\!\cdots\!19}{10\!\cdots\!19}$, $\frac{16822228463965}{56\!\cdots\!01}a^{13}-\frac{15989140692576}{56\!\cdots\!01}a^{12}-\frac{917357032648601}{56\!\cdots\!01}a^{11}+\frac{161217703568104}{56\!\cdots\!01}a^{10}+\frac{16\!\cdots\!60}{56\!\cdots\!01}a^{9}+\frac{38\!\cdots\!44}{56\!\cdots\!01}a^{8}-\frac{12\!\cdots\!61}{56\!\cdots\!01}a^{7}-\frac{58\!\cdots\!46}{56\!\cdots\!01}a^{6}+\frac{36\!\cdots\!78}{56\!\cdots\!01}a^{5}+\frac{21\!\cdots\!98}{56\!\cdots\!01}a^{4}-\frac{13\!\cdots\!76}{56\!\cdots\!01}a^{3}-\frac{10\!\cdots\!65}{56\!\cdots\!01}a^{2}+\frac{23\!\cdots\!09}{56\!\cdots\!01}a-\frac{18\!\cdots\!72}{56\!\cdots\!01}$, $\frac{16\!\cdots\!72}{10\!\cdots\!19}a^{13}-\frac{10\!\cdots\!17}{10\!\cdots\!19}a^{12}-\frac{93\!\cdots\!12}{10\!\cdots\!19}a^{11}-\frac{13\!\cdots\!01}{10\!\cdots\!19}a^{10}+\frac{17\!\cdots\!51}{10\!\cdots\!19}a^{9}+\frac{91\!\cdots\!13}{10\!\cdots\!19}a^{8}-\frac{14\!\cdots\!26}{10\!\cdots\!19}a^{7}-\frac{10\!\cdots\!13}{10\!\cdots\!19}a^{6}+\frac{44\!\cdots\!62}{10\!\cdots\!19}a^{5}+\frac{36\!\cdots\!27}{10\!\cdots\!19}a^{4}-\frac{32\!\cdots\!55}{10\!\cdots\!19}a^{3}-\frac{19\!\cdots\!14}{10\!\cdots\!19}a^{2}+\frac{61\!\cdots\!70}{10\!\cdots\!19}a+\frac{28\!\cdots\!91}{10\!\cdots\!19}$, $\frac{344571995463053}{56\!\cdots\!01}a^{13}-\frac{24\!\cdots\!33}{10\!\cdots\!19}a^{12}-\frac{36\!\cdots\!40}{10\!\cdots\!19}a^{11}-\frac{13\!\cdots\!61}{10\!\cdots\!19}a^{10}+\frac{68\!\cdots\!88}{10\!\cdots\!19}a^{9}+\frac{52\!\cdots\!52}{10\!\cdots\!19}a^{8}-\frac{54\!\cdots\!39}{10\!\cdots\!19}a^{7}-\frac{53\!\cdots\!70}{10\!\cdots\!19}a^{6}+\frac{16\!\cdots\!49}{10\!\cdots\!19}a^{5}+\frac{18\!\cdots\!30}{10\!\cdots\!19}a^{4}-\frac{99\!\cdots\!71}{10\!\cdots\!19}a^{3}-\frac{10\!\cdots\!41}{10\!\cdots\!19}a^{2}+\frac{17\!\cdots\!12}{10\!\cdots\!19}a+\frac{14\!\cdots\!08}{10\!\cdots\!19}$
| 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: | \( 741991346.6880023 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{0}\cdot 741991346.6880023 \cdot 1}{2\cdot\sqrt{401774962552217617093885952}}\cr\approx \mathstrut & 0.303247583959919 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 56*x^12 - 42*x^11 + 1043*x^10 + 1190*x^9 - 7987*x^8 - 11184*x^7 + 22169*x^6 + 37282*x^5 - 5292*x^4 - 22190*x^3 - 3164*x^2 + 3500*x + 881)
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^14 - 56*x^12 - 42*x^11 + 1043*x^10 + 1190*x^9 - 7987*x^8 - 11184*x^7 + 22169*x^6 + 37282*x^5 - 5292*x^4 - 22190*x^3 - 3164*x^2 + 3500*x + 881, 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^14 - 56*x^12 - 42*x^11 + 1043*x^10 + 1190*x^9 - 7987*x^8 - 11184*x^7 + 22169*x^6 + 37282*x^5 - 5292*x^4 - 22190*x^3 - 3164*x^2 + 3500*x + 881);
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^14 - 56*x^12 - 42*x^11 + 1043*x^10 + 1190*x^9 - 7987*x^8 - 11184*x^7 + 22169*x^6 + 37282*x^5 - 5292*x^4 - 22190*x^3 - 3164*x^2 + 3500*x + 881);
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
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 cyclic group of order 14 |
| The 14 conjugacy class representatives for $C_{14}$ |
| Character table for $C_{14}$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), 7.7.13841287201.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.14.0.1}{14} }$ | ${\href{/padicField/5.14.0.1}{14} }$ | R | ${\href{/padicField/11.14.0.1}{14} }$ | ${\href{/padicField/13.14.0.1}{14} }$ | ${\href{/padicField/17.7.0.1}{7} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{7}$ | ${\href{/padicField/23.7.0.1}{7} }^{2}$ | ${\href{/padicField/29.14.0.1}{14} }$ | ${\href{/padicField/31.1.0.1}{1} }^{14}$ | ${\href{/padicField/37.14.0.1}{14} }$ | ${\href{/padicField/41.7.0.1}{7} }^{2}$ | ${\href{/padicField/43.14.0.1}{14} }$ | ${\href{/padicField/47.7.0.1}{7} }^{2}$ | ${\href{/padicField/53.14.0.1}{14} }$ | ${\href{/padicField/59.14.0.1}{14} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.14.21.34 | $x^{14} + 4 x^{13} + 14 x^{12} + 656 x^{11} + 1236 x^{10} - 43472 x^{9} - 244456 x^{8} + 434816 x^{7} + 8570160 x^{6} + 35893184 x^{5} + 77018272 x^{4} + 105671936 x^{3} + 121134528 x^{2} + 74194176 x - 52979584$ | $2$ | $7$ | $21$ | $C_{14}$ | $[3]^{7}$ |
|
\(7\)
| 7.7.12.1 | $x^{7} + 42 x^{6} + 7$ | $7$ | $1$ | $12$ | $C_7$ | $[2]$ |
| 7.7.12.1 | $x^{7} + 42 x^{6} + 7$ | $7$ | $1$ | $12$ | $C_7$ | $[2]$ |