Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 144*x^11 + 161*x^10 + 6530*x^9 - 9620*x^8 - 109398*x^7 + 196143*x^6 + 512628*x^5 - 917970*x^4 - 650724*x^3 + 1134730*x^2 + 253950*x - 409375)
gp: K = bnfinit(y^13 - y^12 - 144*y^11 + 161*y^10 + 6530*y^9 - 9620*y^8 - 109398*y^7 + 196143*y^6 + 512628*y^5 - 917970*y^4 - 650724*y^3 + 1134730*y^2 + 253950*y - 409375, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^13 - x^12 - 144*x^11 + 161*x^10 + 6530*x^9 - 9620*x^8 - 109398*x^7 + 196143*x^6 + 512628*x^5 - 917970*x^4 - 650724*x^3 + 1134730*x^2 + 253950*x - 409375);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^13 - x^12 - 144*x^11 + 161*x^10 + 6530*x^9 - 9620*x^8 - 109398*x^7 + 196143*x^6 + 512628*x^5 - 917970*x^4 - 650724*x^3 + 1134730*x^2 + 253950*x - 409375)
\( x^{13} - x^{12} - 144 x^{11} + 161 x^{10} + 6530 x^{9} - 9620 x^{8} - 109398 x^{7} + 196143 x^{6} + \cdots - 409375 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[13, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(884162417215006648162206715681\)
\(\medspace = 313^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(201.18\) | 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: | $313^{12/13}\approx 201.177491460457$ | ||
| Ramified primes: |
\(313\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $13$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(313\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{313}(1,·)$, $\chi_{313}(294,·)$, $\chi_{313}(103,·)$, $\chi_{313}(234,·)$, $\chi_{313}(44,·)$, $\chi_{313}(48,·)$, $\chi_{313}(113,·)$, $\chi_{313}(277,·)$, $\chi_{313}(150,·)$, $\chi_{313}(280,·)$, $\chi_{313}(249,·)$, $\chi_{313}(58,·)$, $\chi_{313}(27,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{1}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{5}a^{5}-\frac{1}{5}a$, $\frac{1}{5}a^{6}-\frac{1}{5}a^{2}$, $\frac{1}{25}a^{7}+\frac{1}{25}a^{6}+\frac{2}{25}a^{5}+\frac{2}{25}a^{4}-\frac{4}{25}a^{3}+\frac{1}{25}a^{2}-\frac{1}{5}a$, $\frac{1}{25}a^{8}+\frac{1}{25}a^{6}-\frac{1}{25}a^{4}+\frac{2}{5}a^{3}-\frac{1}{25}a^{2}+\frac{2}{5}a$, $\frac{1}{25}a^{9}-\frac{1}{25}a^{6}+\frac{2}{25}a^{5}-\frac{2}{25}a^{4}-\frac{7}{25}a^{3}-\frac{1}{25}a^{2}-\frac{2}{5}a$, $\frac{1}{3625}a^{10}-\frac{7}{725}a^{9}-\frac{16}{3625}a^{8}-\frac{67}{3625}a^{7}-\frac{19}{725}a^{6}+\frac{341}{3625}a^{5}+\frac{172}{3625}a^{4}-\frac{38}{125}a^{3}+\frac{8}{29}a^{2}-\frac{59}{145}a+\frac{6}{29}$, $\frac{1}{2628125}a^{11}-\frac{217}{2628125}a^{10}-\frac{49036}{2628125}a^{9}+\frac{1819}{105125}a^{8}+\frac{15434}{2628125}a^{7}+\frac{166111}{2628125}a^{6}-\frac{4233}{105125}a^{5}+\frac{198289}{2628125}a^{4}+\frac{958754}{2628125}a^{3}+\frac{194174}{525625}a^{2}+\frac{48497}{105125}a+\frac{70}{841}$, $\frac{1}{94\!\cdots\!75}a^{12}-\frac{9218116429}{94\!\cdots\!75}a^{11}+\frac{6077567184843}{94\!\cdots\!75}a^{10}-\frac{217130475748143}{94\!\cdots\!75}a^{9}+\frac{10\!\cdots\!09}{94\!\cdots\!75}a^{8}-\frac{406086214662272}{94\!\cdots\!75}a^{7}-\frac{56\!\cdots\!07}{94\!\cdots\!75}a^{6}-\frac{42\!\cdots\!86}{94\!\cdots\!75}a^{5}+\frac{500807941726986}{94\!\cdots\!75}a^{4}-\frac{22\!\cdots\!78}{94\!\cdots\!75}a^{3}-\frac{95637243330903}{18\!\cdots\!75}a^{2}+\frac{16\!\cdots\!86}{37\!\cdots\!75}a-\frac{781608495677}{30357631559875}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $5$ |
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: | $12$ | 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: |
$\frac{215960315771}{37\!\cdots\!75}a^{12}-\frac{115956091139}{37\!\cdots\!75}a^{11}-\frac{30921180523087}{37\!\cdots\!75}a^{10}+\frac{20294855469427}{37\!\cdots\!75}a^{9}+\frac{13\!\cdots\!14}{37\!\cdots\!75}a^{8}-\frac{14\!\cdots\!82}{37\!\cdots\!75}a^{7}-\frac{22\!\cdots\!02}{37\!\cdots\!75}a^{6}+\frac{30\!\cdots\!94}{37\!\cdots\!75}a^{5}+\frac{10\!\cdots\!36}{37\!\cdots\!75}a^{4}-\frac{11\!\cdots\!33}{37\!\cdots\!75}a^{3}-\frac{19\!\cdots\!83}{758940788996875}a^{2}+\frac{27\!\cdots\!21}{151788157799375}a+\frac{6280796823188}{1214305262395}$, $\frac{23549153895871}{94\!\cdots\!75}a^{12}-\frac{77045675069784}{94\!\cdots\!75}a^{11}-\frac{33\!\cdots\!47}{94\!\cdots\!75}a^{10}+\frac{11\!\cdots\!97}{94\!\cdots\!75}a^{9}+\frac{14\!\cdots\!89}{94\!\cdots\!75}a^{8}-\frac{56\!\cdots\!87}{94\!\cdots\!75}a^{7}-\frac{20\!\cdots\!97}{94\!\cdots\!75}a^{6}+\frac{99\!\cdots\!44}{94\!\cdots\!75}a^{5}+\frac{24\!\cdots\!06}{94\!\cdots\!75}a^{4}-\frac{40\!\cdots\!38}{94\!\cdots\!75}a^{3}+\frac{44\!\cdots\!87}{18\!\cdots\!75}a^{2}+\frac{10\!\cdots\!56}{37\!\cdots\!75}a-\frac{58\!\cdots\!42}{30357631559875}$, $\frac{90267690661916}{94\!\cdots\!75}a^{12}+\frac{21788057017536}{94\!\cdots\!75}a^{11}-\frac{13\!\cdots\!12}{94\!\cdots\!75}a^{10}-\frac{15\!\cdots\!38}{94\!\cdots\!75}a^{9}+\frac{59\!\cdots\!69}{94\!\cdots\!75}a^{8}-\frac{13\!\cdots\!77}{94\!\cdots\!75}a^{7}-\frac{10\!\cdots\!87}{94\!\cdots\!75}a^{6}+\frac{52\!\cdots\!74}{94\!\cdots\!75}a^{5}+\frac{56\!\cdots\!01}{94\!\cdots\!75}a^{4}-\frac{18\!\cdots\!98}{94\!\cdots\!75}a^{3}-\frac{19\!\cdots\!48}{18\!\cdots\!75}a^{2}+\frac{23\!\cdots\!01}{37\!\cdots\!75}a+\frac{13\!\cdots\!18}{30357631559875}$, $\frac{9703097133496}{18\!\cdots\!75}a^{12}-\frac{36093099486684}{18\!\cdots\!75}a^{11}-\frac{13\!\cdots\!97}{18\!\cdots\!75}a^{10}+\frac{53\!\cdots\!97}{18\!\cdots\!75}a^{9}+\frac{58\!\cdots\!89}{18\!\cdots\!75}a^{8}-\frac{26\!\cdots\!37}{18\!\cdots\!75}a^{7}-\frac{77\!\cdots\!72}{18\!\cdots\!75}a^{6}+\frac{45\!\cdots\!44}{18\!\cdots\!75}a^{5}-\frac{51\!\cdots\!94}{18\!\cdots\!75}a^{4}-\frac{18\!\cdots\!38}{18\!\cdots\!75}a^{3}+\frac{31\!\cdots\!87}{37\!\cdots\!75}a^{2}+\frac{55\!\cdots\!31}{758940788996875}a-\frac{39\!\cdots\!17}{6071526311975}$, $\frac{1654472769752}{37\!\cdots\!75}a^{12}-\frac{524484142928}{37\!\cdots\!75}a^{11}-\frac{238060006139749}{37\!\cdots\!75}a^{10}+\frac{103987483467359}{37\!\cdots\!75}a^{9}+\frac{10\!\cdots\!43}{37\!\cdots\!75}a^{8}-\frac{85\!\cdots\!49}{37\!\cdots\!75}a^{7}-\frac{18\!\cdots\!84}{37\!\cdots\!75}a^{6}+\frac{19\!\cdots\!03}{37\!\cdots\!75}a^{5}+\frac{93\!\cdots\!17}{37\!\cdots\!75}a^{4}-\frac{83\!\cdots\!86}{37\!\cdots\!75}a^{3}-\frac{29\!\cdots\!61}{758940788996875}a^{2}+\frac{26\!\cdots\!32}{151788157799375}a+\frac{238087678665311}{1214305262395}$, $\frac{1238104451954}{32\!\cdots\!75}a^{12}-\frac{327792062641}{32\!\cdots\!75}a^{11}-\frac{178479343356028}{32\!\cdots\!75}a^{10}+\frac{68075403667703}{32\!\cdots\!75}a^{9}+\frac{81\!\cdots\!61}{32\!\cdots\!75}a^{8}-\frac{59\!\cdots\!13}{32\!\cdots\!75}a^{7}-\frac{13\!\cdots\!53}{32\!\cdots\!75}a^{6}+\frac{13\!\cdots\!31}{32\!\cdots\!75}a^{5}+\frac{73\!\cdots\!19}{32\!\cdots\!75}a^{4}-\frac{58\!\cdots\!12}{32\!\cdots\!75}a^{3}-\frac{24\!\cdots\!37}{654259300859375}a^{2}+\frac{18\!\cdots\!94}{130851860171875}a+\frac{209211053886767}{1046814881375}$, $\frac{40593472323547}{18\!\cdots\!75}a^{12}-\frac{72156742725638}{18\!\cdots\!75}a^{11}-\frac{57\!\cdots\!54}{18\!\cdots\!75}a^{10}+\frac{11\!\cdots\!29}{18\!\cdots\!75}a^{9}+\frac{25\!\cdots\!73}{18\!\cdots\!75}a^{8}-\frac{58\!\cdots\!09}{18\!\cdots\!75}a^{7}-\frac{39\!\cdots\!54}{18\!\cdots\!75}a^{6}+\frac{10\!\cdots\!83}{18\!\cdots\!75}a^{5}+\frac{11\!\cdots\!42}{18\!\cdots\!75}a^{4}-\frac{44\!\cdots\!41}{18\!\cdots\!75}a^{3}+\frac{23\!\cdots\!59}{37\!\cdots\!75}a^{2}+\frac{12\!\cdots\!92}{758940788996875}a-\frac{48\!\cdots\!94}{6071526311975}$, $\frac{1418076992619}{22\!\cdots\!25}a^{12}-\frac{2564464678551}{22\!\cdots\!25}a^{11}-\frac{201183643291683}{22\!\cdots\!25}a^{10}+\frac{394390630644483}{22\!\cdots\!25}a^{9}+\frac{88\!\cdots\!96}{22\!\cdots\!25}a^{8}-\frac{21\!\cdots\!43}{22\!\cdots\!25}a^{7}-\frac{13\!\cdots\!58}{22\!\cdots\!25}a^{6}+\frac{39\!\cdots\!91}{22\!\cdots\!25}a^{5}+\frac{35\!\cdots\!09}{22\!\cdots\!25}a^{4}-\frac{16\!\cdots\!57}{22\!\cdots\!25}a^{3}+\frac{11\!\cdots\!18}{441244644765625}a^{2}+\frac{44\!\cdots\!34}{88248928953125}a-\frac{208653396110813}{705991431625}$, $\frac{10324326548731}{94\!\cdots\!75}a^{12}-\frac{18925331876724}{94\!\cdots\!75}a^{11}-\frac{14\!\cdots\!92}{94\!\cdots\!75}a^{10}+\frac{29\!\cdots\!67}{94\!\cdots\!75}a^{9}+\frac{61\!\cdots\!54}{94\!\cdots\!75}a^{8}-\frac{15\!\cdots\!07}{94\!\cdots\!75}a^{7}-\frac{86\!\cdots\!17}{94\!\cdots\!75}a^{6}+\frac{28\!\cdots\!09}{94\!\cdots\!75}a^{5}+\frac{92\!\cdots\!41}{94\!\cdots\!75}a^{4}-\frac{11\!\cdots\!43}{94\!\cdots\!75}a^{3}+\frac{15\!\cdots\!57}{18\!\cdots\!75}a^{2}+\frac{29\!\cdots\!16}{37\!\cdots\!75}a-\frac{19\!\cdots\!87}{30357631559875}$, $\frac{30876806594716}{18\!\cdots\!75}a^{12}-\frac{77678038430739}{18\!\cdots\!75}a^{11}-\frac{43\!\cdots\!12}{18\!\cdots\!75}a^{10}+\frac{11\!\cdots\!87}{18\!\cdots\!75}a^{9}+\frac{19\!\cdots\!19}{18\!\cdots\!75}a^{8}-\frac{59\!\cdots\!52}{18\!\cdots\!75}a^{7}-\frac{27\!\cdots\!37}{18\!\cdots\!75}a^{6}+\frac{10\!\cdots\!49}{18\!\cdots\!75}a^{5}+\frac{42\!\cdots\!51}{18\!\cdots\!75}a^{4}-\frac{43\!\cdots\!48}{18\!\cdots\!75}a^{3}+\frac{55\!\cdots\!77}{37\!\cdots\!75}a^{2}+\frac{11\!\cdots\!01}{758940788996875}a-\frac{82\!\cdots\!07}{6071526311975}$, $\frac{228098711291873}{94\!\cdots\!75}a^{12}+\frac{126893751899358}{94\!\cdots\!75}a^{11}-\frac{32\!\cdots\!61}{94\!\cdots\!75}a^{10}-\frac{14\!\cdots\!14}{94\!\cdots\!75}a^{9}+\frac{14\!\cdots\!07}{94\!\cdots\!75}a^{8}+\frac{80\!\cdots\!69}{94\!\cdots\!75}a^{7}-\frac{24\!\cdots\!61}{94\!\cdots\!75}a^{6}+\frac{64\!\cdots\!47}{94\!\cdots\!75}a^{5}+\frac{12\!\cdots\!53}{94\!\cdots\!75}a^{4}-\frac{15\!\cdots\!44}{94\!\cdots\!75}a^{3}-\frac{32\!\cdots\!44}{18\!\cdots\!75}a^{2}+\frac{29\!\cdots\!78}{37\!\cdots\!75}a+\frac{19\!\cdots\!54}{30357631559875}$, $\frac{1582164503863}{18\!\cdots\!75}a^{12}-\frac{47107827577}{18\!\cdots\!75}a^{11}-\frac{226445083439666}{18\!\cdots\!75}a^{10}+\frac{37857484399441}{18\!\cdots\!75}a^{9}+\frac{10\!\cdots\!67}{18\!\cdots\!75}a^{8}-\frac{56\!\cdots\!36}{18\!\cdots\!75}a^{7}-\frac{16\!\cdots\!66}{18\!\cdots\!75}a^{6}+\frac{15\!\cdots\!82}{18\!\cdots\!75}a^{5}+\frac{79\!\cdots\!18}{18\!\cdots\!75}a^{4}-\frac{68\!\cdots\!89}{18\!\cdots\!75}a^{3}-\frac{13\!\cdots\!64}{37\!\cdots\!75}a^{2}+\frac{17\!\cdots\!93}{758940788996875}a+\frac{50277870334724}{6071526311975}$
| 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: | \( 927977436616.0109 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{13}\cdot(2\pi)^{0}\cdot 927977436616.0109 \cdot 1}{2\cdot\sqrt{884162417215006648162206715681}}\cr\approx \mathstrut & 4.04232604063722 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 144*x^11 + 161*x^10 + 6530*x^9 - 9620*x^8 - 109398*x^7 + 196143*x^6 + 512628*x^5 - 917970*x^4 - 650724*x^3 + 1134730*x^2 + 253950*x - 409375)
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^13 - x^12 - 144*x^11 + 161*x^10 + 6530*x^9 - 9620*x^8 - 109398*x^7 + 196143*x^6 + 512628*x^5 - 917970*x^4 - 650724*x^3 + 1134730*x^2 + 253950*x - 409375, 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^13 - x^12 - 144*x^11 + 161*x^10 + 6530*x^9 - 9620*x^8 - 109398*x^7 + 196143*x^6 + 512628*x^5 - 917970*x^4 - 650724*x^3 + 1134730*x^2 + 253950*x - 409375);
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^13 - x^12 - 144*x^11 + 161*x^10 + 6530*x^9 - 9620*x^8 - 109398*x^7 + 196143*x^6 + 512628*x^5 - 917970*x^4 - 650724*x^3 + 1134730*x^2 + 253950*x - 409375);
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 13 |
| The 13 conjugacy class representatives for $C_{13}$ |
| Character table for $C_{13}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 | ${\href{/padicField/2.13.0.1}{13} }$ | ${\href{/padicField/3.13.0.1}{13} }$ | ${\href{/padicField/5.1.0.1}{1} }^{13}$ | ${\href{/padicField/7.13.0.1}{13} }$ | ${\href{/padicField/11.13.0.1}{13} }$ | ${\href{/padicField/13.13.0.1}{13} }$ | ${\href{/padicField/17.13.0.1}{13} }$ | ${\href{/padicField/19.13.0.1}{13} }$ | ${\href{/padicField/23.13.0.1}{13} }$ | ${\href{/padicField/29.1.0.1}{1} }^{13}$ | ${\href{/padicField/31.13.0.1}{13} }$ | ${\href{/padicField/37.13.0.1}{13} }$ | ${\href{/padicField/41.13.0.1}{13} }$ | ${\href{/padicField/43.1.0.1}{1} }^{13}$ | ${\href{/padicField/47.13.0.1}{13} }$ | ${\href{/padicField/53.13.0.1}{13} }$ | ${\href{/padicField/59.13.0.1}{13} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(313\)
| Deg $13$ | $13$ | $1$ | $12$ |