Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 8*x^18 + 4*x^16 + 96*x^14 + 762*x^12 - 160*x^10 - 5176*x^8 + 6288*x^6 + 44361*x^4 + 42280*x^2 + 676)
gp: K = bnfinit(y^20 + 8*y^18 + 4*y^16 + 96*y^14 + 762*y^12 - 160*y^10 - 5176*y^8 + 6288*y^6 + 44361*y^4 + 42280*y^2 + 676, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 8*x^18 + 4*x^16 + 96*x^14 + 762*x^12 - 160*x^10 - 5176*x^8 + 6288*x^6 + 44361*x^4 + 42280*x^2 + 676);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + 8*x^18 + 4*x^16 + 96*x^14 + 762*x^12 - 160*x^10 - 5176*x^8 + 6288*x^6 + 44361*x^4 + 42280*x^2 + 676)
\( x^{20} + 8 x^{18} + 4 x^{16} + 96 x^{14} + 762 x^{12} - 160 x^{10} - 5176 x^{8} + 6288 x^{6} + \cdots + 676 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(6976189580273785130450944000000\)
\(\medspace = 2^{40}\cdot 5^{6}\cdot 67^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(34.85\) | 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: |
\(2\), \(5\), \(67\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{8}a^{11}+\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{3}{8}a^{3}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{8}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{3}{8}a^{4}-\frac{1}{2}$, $\frac{1}{8}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{3}{8}a^{5}-\frac{1}{2}a$, $\frac{1}{8}a^{14}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{3}{8}a^{6}$, $\frac{1}{8}a^{15}-\frac{1}{2}a^{8}-\frac{3}{8}a^{7}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{16}-\frac{1}{2}a^{9}-\frac{3}{8}a^{8}-\frac{1}{2}a^{3}$, $\frac{1}{8}a^{17}-\frac{3}{8}a^{9}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{12\!\cdots\!16}a^{18}-\frac{56760752267087}{12\!\cdots\!16}a^{16}+\frac{15170341227001}{413156147888272}a^{14}+\frac{13291590738321}{413156147888272}a^{12}-\frac{1370051888677}{413156147888272}a^{10}-\frac{1}{2}a^{9}-\frac{238651137966775}{12\!\cdots\!16}a^{8}-\frac{126267527522667}{413156147888272}a^{6}-\frac{153315448412931}{413156147888272}a^{4}+\frac{2945494082151}{103289036972068}a^{2}+\frac{21769298364391}{309867110916204}$, $\frac{1}{16\!\cdots\!08}a^{19}-\frac{676494974099495}{16\!\cdots\!08}a^{17}+\frac{221748415171137}{53\!\cdots\!36}a^{15}-\frac{38352927747713}{53\!\cdots\!36}a^{13}+\frac{205208022055459}{53\!\cdots\!36}a^{11}-\frac{39\!\cdots\!23}{16\!\cdots\!08}a^{9}-\frac{1}{2}a^{8}-\frac{746001749355075}{53\!\cdots\!36}a^{7}-\frac{1}{2}a^{6}+\frac{16\!\cdots\!59}{53\!\cdots\!36}a^{5}-\frac{461855172292155}{13\!\cdots\!84}a^{3}-\frac{1}{2}a^{2}-\frac{133164257093711}{40\!\cdots\!52}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{4}$, which has order $4$ (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: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{5586974899}{33709392819336} a^{19} - \frac{45113426155}{33709392819336} a^{17} - \frac{8638218397}{11236464273112} a^{15} - \frac{179969387501}{11236464273112} a^{13} - \frac{1434380170671}{11236464273112} a^{11} + \frac{512289509689}{33709392819336} a^{9} + \frac{9542111880855}{11236464273112} a^{7} - \frac{11162326717929}{11236464273112} a^{5} - \frac{41792532686317}{5618232136556} a^{3} - \frac{33547646840519}{4213674102417} a \)
(order $4$)
| 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: |
$\frac{29204708521457}{53\!\cdots\!36}a^{19}-\frac{1356983247755}{103289036972068}a^{18}+\frac{164843515967671}{53\!\cdots\!36}a^{17}-\frac{7361529180943}{103289036972068}a^{16}-\frac{279316939960535}{53\!\cdots\!36}a^{15}+\frac{13606653462063}{103289036972068}a^{14}+\frac{34\!\cdots\!15}{53\!\cdots\!36}a^{13}-\frac{165041011431361}{103289036972068}a^{12}+\frac{14\!\cdots\!07}{53\!\cdots\!36}a^{11}-\frac{610222344323461}{103289036972068}a^{10}-\frac{39\!\cdots\!33}{53\!\cdots\!36}a^{9}+\frac{18\!\cdots\!29}{103289036972068}a^{8}-\frac{61\!\cdots\!35}{53\!\cdots\!36}a^{7}+\frac{23\!\cdots\!21}{103289036972068}a^{6}+\frac{33\!\cdots\!19}{53\!\cdots\!36}a^{5}-\frac{14\!\cdots\!25}{103289036972068}a^{4}+\frac{25\!\cdots\!51}{26\!\cdots\!68}a^{3}-\frac{54\!\cdots\!05}{25822259243017}a^{2}-\frac{19\!\cdots\!63}{335689370159221}a-\frac{192446254098553}{51644518486034}$, $\frac{1655077336465}{53\!\cdots\!36}a^{19}+\frac{3481596049957}{619734221832408}a^{18}+\frac{6960216220403}{53\!\cdots\!36}a^{17}+\frac{2501754393209}{77466777729051}a^{16}-\frac{18854329776345}{53\!\cdots\!36}a^{15}-\frac{10421783501607}{206578073944136}a^{14}+\frac{252314721648531}{53\!\cdots\!36}a^{13}+\frac{16836680334531}{25822259243017}a^{12}+\frac{357594713085445}{53\!\cdots\!36}a^{11}+\frac{581242491912911}{206578073944136}a^{10}-\frac{17\!\cdots\!85}{53\!\cdots\!36}a^{9}-\frac{563812737218066}{77466777729051}a^{8}+\frac{11\!\cdots\!51}{53\!\cdots\!36}a^{7}-\frac{26\!\cdots\!31}{206578073944136}a^{6}+\frac{79\!\cdots\!55}{53\!\cdots\!36}a^{5}+\frac{33\!\cdots\!89}{51644518486034}a^{4}+\frac{11\!\cdots\!33}{335689370159221}a^{3}+\frac{53\!\cdots\!69}{51644518486034}a^{2}+\frac{95\!\cdots\!45}{13\!\cdots\!84}a+\frac{163964562582410}{77466777729051}$, $\frac{657613200486}{335689370159221}a^{19}+\frac{3379159082813}{619734221832408}a^{18}+\frac{27763206050507}{26\!\cdots\!68}a^{17}+\frac{9523086549163}{309867110916204}a^{16}-\frac{28902571081789}{13\!\cdots\!84}a^{15}-\frac{10557280086293}{206578073944136}a^{14}+\frac{80567696271310}{335689370159221}a^{13}+\frac{16604480959152}{25822259243017}a^{12}+\frac{22\!\cdots\!27}{26\!\cdots\!68}a^{11}+\frac{545097264999673}{206578073944136}a^{10}-\frac{74\!\cdots\!39}{26\!\cdots\!68}a^{9}-\frac{22\!\cdots\!01}{309867110916204}a^{8}-\frac{41\!\cdots\!95}{13\!\cdots\!84}a^{7}-\frac{23\!\cdots\!09}{206578073944136}a^{6}+\frac{15\!\cdots\!75}{671378740318442}a^{5}+\frac{15\!\cdots\!45}{25822259243017}a^{4}+\frac{75\!\cdots\!47}{26\!\cdots\!68}a^{3}+\frac{99\!\cdots\!61}{103289036972068}a^{2}-\frac{94\!\cdots\!19}{13\!\cdots\!84}a+\frac{96809375258737}{77466777729051}$, $\frac{15521732561917}{53\!\cdots\!36}a^{19}-\frac{513084121233}{206578073944136}a^{18}+\frac{103172341725091}{53\!\cdots\!36}a^{17}-\frac{1262690849089}{103289036972068}a^{16}-\frac{73820261181445}{53\!\cdots\!36}a^{15}+\frac{5966783079055}{206578073944136}a^{14}+\frac{16\!\cdots\!09}{53\!\cdots\!36}a^{13}-\frac{16769594735107}{51644518486034}a^{12}+\frac{96\!\cdots\!55}{53\!\cdots\!36}a^{11}-\frac{189816568742897}{206578073944136}a^{10}-\frac{15\!\cdots\!61}{53\!\cdots\!36}a^{9}+\frac{354671945211975}{103289036972068}a^{8}-\frac{55\!\cdots\!29}{53\!\cdots\!36}a^{7}+\frac{463316621352331}{206578073944136}a^{6}+\frac{16\!\cdots\!21}{53\!\cdots\!36}a^{5}-\frac{12\!\cdots\!73}{51644518486034}a^{4}+\frac{21\!\cdots\!63}{26\!\cdots\!68}a^{3}-\frac{15\!\cdots\!45}{51644518486034}a^{2}+\frac{18\!\cdots\!97}{671378740318442}a-\frac{256870859102083}{51644518486034}$, $\frac{1651175057075}{103289036972068}a^{18}+\frac{4546987940957}{51644518486034}a^{16}-\frac{16091310749783}{103289036972068}a^{14}+\frac{198519706934561}{103289036972068}a^{12}+\frac{763941685047527}{103289036972068}a^{10}-\frac{10\!\cdots\!23}{51644518486034}a^{8}-\frac{31\!\cdots\!51}{103289036972068}a^{6}+\frac{18\!\cdots\!81}{103289036972068}a^{4}+\frac{69\!\cdots\!07}{25822259243017}a^{2}+\frac{97669147841540}{25822259243017}$, $\frac{22016983388611}{53\!\cdots\!36}a^{19}+\frac{815427199659}{206578073944136}a^{18}+\frac{120513728590877}{53\!\cdots\!36}a^{17}+\frac{1149946956229}{51644518486034}a^{16}-\frac{219704495890153}{53\!\cdots\!36}a^{15}-\frac{7589944349401}{206578073944136}a^{14}+\frac{26\!\cdots\!61}{53\!\cdots\!36}a^{13}+\frac{48026532689483}{103289036972068}a^{12}+\frac{10\!\cdots\!61}{53\!\cdots\!36}a^{11}+\frac{393845779155351}{206578073944136}a^{10}-\frac{29\!\cdots\!79}{53\!\cdots\!36}a^{9}-\frac{264886187613327}{51644518486034}a^{8}-\frac{40\!\cdots\!41}{53\!\cdots\!36}a^{7}-\frac{17\!\cdots\!53}{206578073944136}a^{6}+\frac{24\!\cdots\!57}{53\!\cdots\!36}a^{5}+\frac{45\!\cdots\!87}{103289036972068}a^{4}+\frac{17\!\cdots\!53}{26\!\cdots\!68}a^{3}+\frac{36\!\cdots\!15}{51644518486034}a^{2}-\frac{931704896303450}{335689370159221}a+\frac{18341074344075}{51644518486034}$, $\frac{14907514311455}{80\!\cdots\!04}a^{19}-\frac{173813865475}{154933555458102}a^{18}+\frac{55726078122217}{40\!\cdots\!52}a^{17}-\frac{2961309309307}{619734221832408}a^{16}+\frac{236208727991}{335689370159221}a^{15}+\frac{428362665785}{25822259243017}a^{14}+\frac{245535213463927}{13\!\cdots\!84}a^{13}-\frac{8007003173087}{51644518486034}a^{12}+\frac{441978323506061}{335689370159221}a^{11}-\frac{16285562501907}{51644518486034}a^{10}-\frac{970009782996247}{10\!\cdots\!63}a^{9}+\frac{11\!\cdots\!45}{619734221832408}a^{8}-\frac{28\!\cdots\!48}{335689370159221}a^{7}+\frac{2934501468450}{25822259243017}a^{6}+\frac{20\!\cdots\!39}{13\!\cdots\!84}a^{5}-\frac{617990103393593}{51644518486034}a^{4}+\frac{18\!\cdots\!39}{26\!\cdots\!68}a^{3}-\frac{178007751720622}{25822259243017}a^{2}+\frac{21\!\cdots\!85}{40\!\cdots\!52}a+\frac{13\!\cdots\!89}{154933555458102}$, $\frac{74534032909813}{80\!\cdots\!04}a^{19}+\frac{8365068877235}{154933555458102}a^{18}+\frac{419764555852753}{80\!\cdots\!04}a^{17}+\frac{185493531868229}{619734221832408}a^{16}-\frac{224970112121431}{26\!\cdots\!68}a^{15}-\frac{26799809006797}{51644518486034}a^{14}+\frac{29\!\cdots\!27}{26\!\cdots\!68}a^{13}+\frac{13\!\cdots\!31}{206578073944136}a^{12}+\frac{29\!\cdots\!55}{671378740318442}a^{11}+\frac{13\!\cdots\!99}{51644518486034}a^{10}-\frac{93\!\cdots\!17}{80\!\cdots\!04}a^{9}-\frac{43\!\cdots\!31}{619734221832408}a^{8}-\frac{51\!\cdots\!23}{26\!\cdots\!68}a^{7}-\frac{27\!\cdots\!24}{25822259243017}a^{6}+\frac{26\!\cdots\!63}{26\!\cdots\!68}a^{5}+\frac{12\!\cdots\!75}{206578073944136}a^{4}+\frac{45\!\cdots\!01}{26\!\cdots\!68}a^{3}+\frac{23\!\cdots\!06}{25822259243017}a^{2}+\frac{88\!\cdots\!13}{40\!\cdots\!52}a+\frac{12\!\cdots\!81}{77466777729051}$, $\frac{70274469908165}{53\!\cdots\!36}a^{19}+\frac{22421865558803}{309867110916204}a^{18}+\frac{387393092577073}{53\!\cdots\!36}a^{17}+\frac{249734505798691}{619734221832408}a^{16}-\frac{682694709487571}{53\!\cdots\!36}a^{15}-\frac{17823718387245}{25822259243017}a^{14}+\frac{84\!\cdots\!55}{53\!\cdots\!36}a^{13}+\frac{890801798513505}{103289036972068}a^{12}+\frac{32\!\cdots\!89}{53\!\cdots\!36}a^{11}+\frac{35\!\cdots\!81}{103289036972068}a^{10}-\frac{92\!\cdots\!19}{53\!\cdots\!36}a^{9}-\frac{58\!\cdots\!29}{619734221832408}a^{8}-\frac{13\!\cdots\!59}{53\!\cdots\!36}a^{7}-\frac{37\!\cdots\!24}{25822259243017}a^{6}+\frac{77\!\cdots\!27}{53\!\cdots\!36}a^{5}+\frac{83\!\cdots\!85}{103289036972068}a^{4}+\frac{14\!\cdots\!13}{671378740318442}a^{3}+\frac{32\!\cdots\!97}{25822259243017}a^{2}+\frac{62\!\cdots\!33}{13\!\cdots\!84}a+\frac{15\!\cdots\!13}{77466777729051}$
| 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: | \( 22547882.1224 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 22547882.1224 \cdot 4}{4\cdot\sqrt{6976189580273785130450944000000}}\cr\approx \mathstrut & 0.818644412932 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 + 8*x^18 + 4*x^16 + 96*x^14 + 762*x^12 - 160*x^10 - 5176*x^8 + 6288*x^6 + 44361*x^4 + 42280*x^2 + 676)
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^20 + 8*x^18 + 4*x^16 + 96*x^14 + 762*x^12 - 160*x^10 - 5176*x^8 + 6288*x^6 + 44361*x^4 + 42280*x^2 + 676, 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^20 + 8*x^18 + 4*x^16 + 96*x^14 + 762*x^12 - 160*x^10 - 5176*x^8 + 6288*x^6 + 44361*x^4 + 42280*x^2 + 676);
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^20 + 8*x^18 + 4*x^16 + 96*x^14 + 762*x^12 - 160*x^10 - 5176*x^8 + 6288*x^6 + 44361*x^4 + 42280*x^2 + 676);
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
$C_2\wr S_5$ (as 20T288):
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 non-solvable group of order 3840 |
| The 36 conjugacy class representatives for $C_2\wr S_5$ |
| Character table for $C_2\wr S_5$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 5.5.5745920.1, 10.0.528249546342400.1, 10.4.2641247731712000.1, 10.6.165077983232000.1 |
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 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.4.2641247731712000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
| Deg $16$ | $8$ | $2$ | $36$ | ||||
|
\(5\)
| 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
|
\(67\)
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.6.4.1 | $x^{6} + 189 x^{5} + 11913 x^{4} + 250937 x^{3} + 36489 x^{2} + 797721 x + 16732320$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 67.6.4.1 | $x^{6} + 189 x^{5} + 11913 x^{4} + 250937 x^{3} + 36489 x^{2} + 797721 x + 16732320$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |