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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 274170.ck
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 274170.ck1 | 274170ck8 | \([1, 0, 0, -144530433645, -21148877394690513]\) | \(333888629342126551846198224694373785681/10585725119531250\) | \(10585725119531250\) | \([2]\) | \(668860416\) | \(4.4993\) | |
| 274170.ck2 | 274170ck6 | \([1, 0, 0, -9033152115, -330451772926275]\) | \(81515778978451355139317145508643761/458987832550501978522500\) | \(458987832550501978522500\) | \([2, 2]\) | \(334430208\) | \(4.1527\) | |
| 274170.ck3 | 274170ck7 | \([1, 0, 0, -9028007865, -330846942893925]\) | \(-81376592087990245278753761880191761/193439944959886548255650492850\) | \(-193439944959886548255650492850\) | \([2]\) | \(668860416\) | \(4.4993\) | |
| 274170.ck4 | 274170ck5 | \([1, 0, 0, -1784339895, -29010475159263]\) | \(628284678908027418152998936285681/19884708881378173828125000\) | \(19884708881378173828125000\) | \([6]\) | \(222953472\) | \(3.9500\) | |
| 274170.ck5 | 274170ck3 | \([1, 0, 0, -564893535, -5157168789303]\) | \(19935334419592113902235231408241/47221170298807016219619600\) | \(47221170298807016219619600\) | \([4]\) | \(167215104\) | \(3.8062\) | |
| 274170.ck6 | 274170ck2 | \([1, 0, 0, -116289615, -412420328775]\) | \(173918792090461264622472443761/27171794302813265625000000\) | \(27171794302813265625000000\) | \([2, 6]\) | \(111476736\) | \(3.6034\) | |
| 274170.ck7 | 274170ck1 | \([1, 0, 0, -32075535, 63675551097]\) | \(3649601442456329976279696241/360314065091912256000000\) | \(360314065091912256000000\) | \([12]\) | \(55738368\) | \(3.2568\) | \(\Gamma_0(N)\)-optimal |
| 274170.ck8 | 274170ck4 | \([1, 0, 0, 204335385, -2284164953775]\) | \(943527262510267812948497556239/2799977246032887524404125000\) | \(-2799977246032887524404125000\) | \([6]\) | \(222953472\) | \(3.9500\) |
Rank
sage: E.rank()
The elliptic curves in class 274170.ck have rank \(1\).
Complex multiplication
The elliptic curves in class 274170.ck do not have complex multiplication.Modular form 274170.2.a.ck
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
