Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 75*x^18 - 390*x^17 + 1602*x^16 - 5268*x^15 + 14090*x^14 - 30920*x^13 + 55743*x^12 - 82466*x^11 + 99350*x^10 - 96015*x^9 + 72432*x^8 - 40386*x^7 + 14757*x^6 - 2239*x^5 - 461*x^4 - 84*x^3 + 336*x^2 - 147*x + 21)
gp: K = bnfinit(y^20 - 10*y^19 + 75*y^18 - 390*y^17 + 1602*y^16 - 5268*y^15 + 14090*y^14 - 30920*y^13 + 55743*y^12 - 82466*y^11 + 99350*y^10 - 96015*y^9 + 72432*y^8 - 40386*y^7 + 14757*y^6 - 2239*y^5 - 461*y^4 - 84*y^3 + 336*y^2 - 147*y + 21, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 10*x^19 + 75*x^18 - 390*x^17 + 1602*x^16 - 5268*x^15 + 14090*x^14 - 30920*x^13 + 55743*x^12 - 82466*x^11 + 99350*x^10 - 96015*x^9 + 72432*x^8 - 40386*x^7 + 14757*x^6 - 2239*x^5 - 461*x^4 - 84*x^3 + 336*x^2 - 147*x + 21);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 10*x^19 + 75*x^18 - 390*x^17 + 1602*x^16 - 5268*x^15 + 14090*x^14 - 30920*x^13 + 55743*x^12 - 82466*x^11 + 99350*x^10 - 96015*x^9 + 72432*x^8 - 40386*x^7 + 14757*x^6 - 2239*x^5 - 461*x^4 - 84*x^3 + 336*x^2 - 147*x + 21)
\( x^{20} - 10 x^{19} + 75 x^{18} - 390 x^{17} + 1602 x^{16} - 5268 x^{15} + 14090 x^{14} - 30920 x^{13} + \cdots + 21 \)
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: |
\(654253297161106870378547679969\)
\(\medspace = 3^{28}\cdot 7^{6}\cdot 79^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(30.96\) | 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: | $3^{23/12}7^{3/4}79^{3/4}\approx 936.5398325690629$ | ||
| Ramified primes: |
\(3\), \(7\), \(79\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{16}+\frac{1}{3}a^{10}+\frac{1}{3}a^{4}$, $\frac{1}{3}a^{17}+\frac{1}{3}a^{11}+\frac{1}{3}a^{5}$, $\frac{1}{4424737629}a^{18}-\frac{3}{1474912543}a^{17}-\frac{345077830}{4424737629}a^{16}-\frac{63067414}{1474912543}a^{15}+\frac{177722150}{1474912543}a^{14}-\frac{140375543}{1474912543}a^{13}-\frac{840658613}{4424737629}a^{12}+\frac{636632832}{1474912543}a^{11}+\frac{1365593618}{4424737629}a^{10}+\frac{237042511}{1474912543}a^{9}+\frac{246164580}{1474912543}a^{8}-\frac{246739417}{1474912543}a^{7}+\frac{284095381}{4424737629}a^{6}-\frac{641958939}{1474912543}a^{5}+\frac{1735667189}{4424737629}a^{4}-\frac{340344575}{1474912543}a^{3}-\frac{696368617}{1474912543}a^{2}+\frac{98085853}{1474912543}a+\frac{676125672}{1474912543}$, $\frac{1}{4424737629}a^{19}-\frac{345077911}{4424737629}a^{17}-\frac{345077626}{4424737629}a^{16}+\frac{305258815}{4424737629}a^{15}-\frac{15788736}{1474912543}a^{14}-\frac{560324396}{1474912543}a^{13}+\frac{243621151}{4424737629}a^{12}+\frac{855729566}{4424737629}a^{11}+\frac{1202169751}{4424737629}a^{10}+\frac{1238991365}{4424737629}a^{9}+\frac{493829260}{1474912543}a^{8}-\frac{478218706}{4424737629}a^{7}-\frac{843930931}{4424737629}a^{6}+\frac{2101726352}{4424737629}a^{5}+\frac{441919363}{1474912543}a^{4}+\frac{51174449}{113454811}a^{3}-\frac{269581528}{1474912543}a^{2}+\frac{83985806}{1474912543}a+\frac{185480876}{1474912543}$
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
$C_{2}\times C_{4}$, which has order $8$ (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: |
\( \frac{35095174}{414573} a^{19} - \frac{333404153}{414573} a^{18} + \frac{821808535}{138191} a^{17} - \frac{4151437247}{138191} a^{16} + \frac{49994646140}{414573} a^{15} - \frac{53293611612}{138191} a^{14} + \frac{414538330490}{414573} a^{13} - \frac{292612254389}{138191} a^{12} + \frac{505767948756}{138191} a^{11} - \frac{2135338474280}{414573} a^{10} + \frac{2418782525449}{414573} a^{9} - \frac{719991559532}{138191} a^{8} + \frac{487255237592}{138191} a^{7} - \frac{686313903214}{414573} a^{6} + \frac{58230323547}{138191} a^{5} + \frac{8765386439}{414573} a^{4} - \frac{3930071607}{138191} a^{3} - \frac{2946519135}{138191} a^{2} + \frac{2456270922}{138191} a - \frac{491365827}{138191} \)
(order $6$)
| 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: |
$\frac{344219473001}{4424737629}a^{19}-\frac{1090147749153}{1474912543}a^{18}+\frac{8061489527966}{1474912543}a^{17}-\frac{122176853936302}{4424737629}a^{16}+\frac{163488702103138}{1474912543}a^{15}-\frac{15\!\cdots\!82}{4424737629}a^{14}+\frac{312858508823395}{340364433}a^{13}-\frac{28\!\cdots\!16}{1474912543}a^{12}+\frac{14\!\cdots\!66}{4424737629}a^{11}-\frac{20\!\cdots\!63}{4424737629}a^{10}+\frac{79\!\cdots\!67}{1474912543}a^{9}-\frac{16\!\cdots\!97}{340364433}a^{8}+\frac{47\!\cdots\!12}{1474912543}a^{7}-\frac{22\!\cdots\!54}{1474912543}a^{6}+\frac{17\!\cdots\!94}{4424737629}a^{5}+\frac{28609552829424}{1474912543}a^{4}-\frac{38571396876179}{1474912543}a^{3}-\frac{2224517664053}{113454811}a^{2}+\frac{24106883710921}{1474912543}a-\frac{4822469757250}{1474912543}$, $\frac{194810208859}{4424737629}a^{19}-\frac{616928782343}{1474912543}a^{18}+\frac{4562060801905}{1474912543}a^{17}-\frac{23046284034529}{1474912543}a^{16}+\frac{277546089463817}{4424737629}a^{15}-\frac{887604915353785}{4424737629}a^{14}+\frac{767149202787260}{1474912543}a^{13}-\frac{48\!\cdots\!00}{4424737629}a^{12}+\frac{84\!\cdots\!84}{4424737629}a^{11}-\frac{11\!\cdots\!49}{4424737629}a^{10}+\frac{13\!\cdots\!73}{4424737629}a^{9}-\frac{11\!\cdots\!66}{4424737629}a^{8}+\frac{81\!\cdots\!03}{4424737629}a^{7}-\frac{38\!\cdots\!16}{4424737629}a^{6}+\frac{971067218316220}{4424737629}a^{5}+\frac{48397350832435}{4424737629}a^{4}-\frac{21839641168211}{1474912543}a^{3}-\frac{16349847852227}{1474912543}a^{2}+\frac{13647369287684}{1474912543}a-\frac{2732564233900}{1474912543}$, $\frac{59681358448}{1474912543}a^{19}-\frac{1701187606849}{4424737629}a^{18}+\frac{12580415606854}{4424737629}a^{17}-\frac{63557068962734}{4424737629}a^{16}+\frac{255153925479337}{4424737629}a^{15}-\frac{816048337591505}{4424737629}a^{14}+\frac{21\!\cdots\!53}{4424737629}a^{13}-\frac{14\!\cdots\!95}{1474912543}a^{12}+\frac{77\!\cdots\!26}{4424737629}a^{11}-\frac{10\!\cdots\!60}{4424737629}a^{10}+\frac{12\!\cdots\!75}{4424737629}a^{9}-\frac{11\!\cdots\!16}{4424737629}a^{8}+\frac{74\!\cdots\!31}{4424737629}a^{7}-\frac{35\!\cdots\!27}{4424737629}a^{6}+\frac{22901876426668}{113454811}a^{5}+\frac{14915794547783}{1474912543}a^{4}-\frac{20111189854174}{1474912543}a^{3}-\frac{15053391798242}{1474912543}a^{2}+\frac{12554928235455}{1474912543}a-\frac{2511783237671}{1474912543}$, $\frac{171751482551}{4424737629}a^{19}-\frac{1631465003029}{4424737629}a^{18}+\frac{12063998540756}{4424737629}a^{17}-\frac{60939348904231}{4424737629}a^{16}+\frac{81539465313974}{1474912543}a^{15}-\frac{260751488960122}{1474912543}a^{14}+\frac{20\!\cdots\!97}{4424737629}a^{13}-\frac{42\!\cdots\!72}{4424737629}a^{12}+\frac{74\!\cdots\!32}{4424737629}a^{11}-\frac{10\!\cdots\!13}{4424737629}a^{10}+\frac{39\!\cdots\!50}{1474912543}a^{9}-\frac{35\!\cdots\!38}{1474912543}a^{8}+\frac{71\!\cdots\!41}{4424737629}a^{7}-\frac{33\!\cdots\!29}{4424737629}a^{6}+\frac{855725932144529}{4424737629}a^{5}+\frac{42495997921673}{4424737629}a^{4}-\frac{19215205699772}{1474912543}a^{3}-\frac{14422079619735}{1474912543}a^{2}+\frac{12025271738791}{1474912543}a-\frac{185099866835}{113454811}$, $\frac{795364020788}{4424737629}a^{19}-\frac{7556486207662}{4424737629}a^{18}+\frac{18626323014685}{1474912543}a^{17}-\frac{94096155612248}{1474912543}a^{16}+\frac{11\!\cdots\!36}{4424737629}a^{15}-\frac{12\!\cdots\!19}{1474912543}a^{14}+\frac{722836089171862}{340364433}a^{13}-\frac{66\!\cdots\!34}{1474912543}a^{12}+\frac{11\!\cdots\!70}{1474912543}a^{11}-\frac{48\!\cdots\!19}{4424737629}a^{10}+\frac{54\!\cdots\!73}{4424737629}a^{9}-\frac{12\!\cdots\!18}{113454811}a^{8}+\frac{11\!\cdots\!67}{1474912543}a^{7}-\frac{15\!\cdots\!85}{4424737629}a^{6}+\frac{13\!\cdots\!02}{1474912543}a^{5}+\frac{198492858055078}{4424737629}a^{4}-\frac{89136998082537}{1474912543}a^{3}-\frac{5139965384542}{113454811}a^{2}+\frac{55705762368375}{1474912543}a-\frac{11145026890125}{1474912543}$, $\frac{10117106367}{1474912543}a^{19}-\frac{288695785409}{4424737629}a^{18}+\frac{711822489786}{1474912543}a^{17}-\frac{10794904744987}{4424737629}a^{16}+\frac{43355700088178}{4424737629}a^{15}-\frac{138736844775812}{4424737629}a^{14}+\frac{119987294624991}{1474912543}a^{13}-\frac{254280706465460}{1474912543}a^{12}+\frac{13\!\cdots\!04}{4424737629}a^{11}-\frac{18\!\cdots\!31}{4424737629}a^{10}+\frac{21\!\cdots\!07}{4424737629}a^{9}-\frac{18\!\cdots\!76}{4424737629}a^{8}+\frac{425993600392146}{1474912543}a^{7}-\frac{601499737262974}{4424737629}a^{6}+\frac{153797357444378}{4424737629}a^{5}+\frac{7487980280054}{4424737629}a^{4}-\frac{3510335714553}{1474912543}a^{3}-\frac{2534645856040}{1474912543}a^{2}+\frac{2153854916971}{1474912543}a-\frac{435359741688}{1474912543}$, $\frac{343330568522}{4424737629}a^{19}-\frac{1087184734223}{1474912543}a^{18}+\frac{24117962250539}{4424737629}a^{17}-\frac{121831634712586}{4424737629}a^{16}+\frac{489052771142954}{4424737629}a^{15}-\frac{521311323963410}{1474912543}a^{14}+\frac{13\!\cdots\!39}{1474912543}a^{13}-\frac{85\!\cdots\!36}{4424737629}a^{12}+\frac{14\!\cdots\!02}{4424737629}a^{11}-\frac{69\!\cdots\!97}{1474912543}a^{10}+\frac{23\!\cdots\!73}{4424737629}a^{9}-\frac{70\!\cdots\!81}{1474912543}a^{8}+\frac{14\!\cdots\!02}{4424737629}a^{7}-\frac{67\!\cdots\!20}{4424737629}a^{6}+\frac{17\!\cdots\!02}{4424737629}a^{5}+\frac{86195468027923}{4424737629}a^{4}-\frac{38374556265782}{1474912543}a^{3}-\frac{28854721039202}{1474912543}a^{2}+\frac{24013535470108}{1474912543}a-\frac{4799368263294}{1474912543}$, $\frac{3088756223}{4424737629}a^{18}-\frac{9266268669}{1474912543}a^{17}+\frac{67532147123}{1474912543}a^{16}-\frac{330221753820}{1474912543}a^{15}+\frac{1291713657511}{1474912543}a^{14}-\frac{3998238891801}{1474912543}a^{13}+\frac{29963572262012}{4424737629}a^{12}-\frac{20244668228289}{1474912543}a^{11}+\frac{33180689891221}{1474912543}a^{10}-\frac{43776024999276}{1474912543}a^{9}+\frac{45679062260318}{1474912543}a^{8}-\frac{36575095666227}{1474912543}a^{7}+\frac{63329054264345}{4424737629}a^{6}-\frac{7396621123176}{1474912543}a^{5}+\frac{592064493925}{1474912543}a^{4}+\frac{628213167966}{1474912543}a^{3}+\frac{33726707484}{1474912543}a^{2}-\frac{241437155150}{1474912543}a+\frac{72138209891}{1474912543}$, $\frac{984210250}{1474912543}a^{18}-\frac{8857892250}{1474912543}a^{17}+\frac{193184577874}{4424737629}a^{16}-\frac{943139949992}{4424737629}a^{15}+\frac{3677072381039}{4424737629}a^{14}-\frac{3780911965971}{1474912543}a^{13}+\frac{28190789788777}{4424737629}a^{12}-\frac{56771663035429}{4424737629}a^{11}+\frac{92261994878101}{4424737629}a^{10}-\frac{120380960818369}{4424737629}a^{9}+\frac{123856566372670}{4424737629}a^{8}-\frac{32449549508865}{1474912543}a^{7}+\frac{54819071210519}{4424737629}a^{6}-\frac{18509181337493}{4424737629}a^{5}+\frac{1269278497978}{4424737629}a^{4}+\frac{492402443251}{1474912543}a^{3}+\frac{57985215928}{1474912543}a^{2}-\frac{199723357568}{1474912543}a+\frac{61037952876}{1474912543}$
| 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: | \( 4740283.57365 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 4740283.57365 \cdot 8}{6\cdot\sqrt{654253297161106870378547679969}}\cr\approx \mathstrut & 0.749322516641 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 75*x^18 - 390*x^17 + 1602*x^16 - 5268*x^15 + 14090*x^14 - 30920*x^13 + 55743*x^12 - 82466*x^11 + 99350*x^10 - 96015*x^9 + 72432*x^8 - 40386*x^7 + 14757*x^6 - 2239*x^5 - 461*x^4 - 84*x^3 + 336*x^2 - 147*x + 21)
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 - 10*x^19 + 75*x^18 - 390*x^17 + 1602*x^16 - 5268*x^15 + 14090*x^14 - 30920*x^13 + 55743*x^12 - 82466*x^11 + 99350*x^10 - 96015*x^9 + 72432*x^8 - 40386*x^7 + 14757*x^6 - 2239*x^5 - 461*x^4 - 84*x^3 + 336*x^2 - 147*x + 21, 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 - 10*x^19 + 75*x^18 - 390*x^17 + 1602*x^16 - 5268*x^15 + 14090*x^14 - 30920*x^13 + 55743*x^12 - 82466*x^11 + 99350*x^10 - 96015*x^9 + 72432*x^8 - 40386*x^7 + 14757*x^6 - 2239*x^5 - 461*x^4 - 84*x^3 + 336*x^2 - 147*x + 21);
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 - 10*x^19 + 75*x^18 - 390*x^17 + 1602*x^16 - 5268*x^15 + 14090*x^14 - 30920*x^13 + 55743*x^12 - 82466*x^11 + 99350*x^10 - 96015*x^9 + 72432*x^8 - 40386*x^7 + 14757*x^6 - 2239*x^5 - 461*x^4 - 84*x^3 + 336*x^2 - 147*x + 21);
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 $C_2\wr S_5$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 5.5.403137.1, 10.0.487558322307.1, 10.8.269619752235771.1, 10.2.808859256707313.1 |
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: | 10.8.269619752235771.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/5.8.0.1}{8} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/11.10.0.1}{10} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.10.0.1}{10} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{10}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{10}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
| 3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 3.12.22.44 | $x^{12} + 105 x^{6} + 144$ | $6$ | $2$ | $22$ | $D_6$ | $[5/2]_{2}^{2}$ | |
|
\(7\)
| 7.4.3.2 | $x^{4} + 21$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
| 7.4.3.2 | $x^{4} + 21$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 7.6.0.1 | $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 7.6.0.1 | $x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(79\)
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.4.3.2 | $x^{4} + 237$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 79.4.0.1 | $x^{4} + 2 x^{2} + 66 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 79.4.0.1 | $x^{4} + 2 x^{2} + 66 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 79.4.3.2 | $x^{4} + 237$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |