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SageMath
E = EllipticCurve("ep1")
E.isogeny_class()
Elliptic curves in class 430950ep
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 430950.ep7 | 430950ep1 | \([1, 1, 1, -8856216938, 308890597831031]\) | \(1018563973439611524445729/42904970360310988800\) | \(3235845266873161305292800000000\) | \([2]\) | \(1114767360\) | \(4.6189\) | \(\Gamma_0(N)\)-optimal |
| 430950.ep6 | 430950ep2 | \([1, 1, 1, -23479448938, -974034791992969]\) | \(18980483520595353274840609/5549773448629762560000\) | \(418557756715737118632360000000000\) | \([2, 2]\) | \(2229534720\) | \(4.9655\) | |
| 430950.ep5 | 430950ep3 | \([1, 1, 1, -108974520938, -13756221892792969]\) | \(1897660325010178513043539489/14258428094958372000000\) | \(1075354828978092571796062500000000\) | \([2]\) | \(3344302080\) | \(5.1682\) | |
| 430950.ep8 | 430950ep4 | \([1, 1, 1, 62871439062, -6473895550488969]\) | \(364421318680576777174674911/450962301637624725000000\) | \(-34011076503206277519570703125000000\) | \([2]\) | \(4459069440\) | \(5.3121\) | |
| 430950.ep4 | 430950ep5 | \([1, 1, 1, -343802048938, -77581106517592969]\) | \(59589391972023341137821784609/8834417507562311995200\) | \(666281967738427118738114325000000\) | \([2]\) | \(4459069440\) | \(5.3121\) | |
| 430950.ep2 | 430950ep6 | \([1, 1, 1, -1740435962938, -883762402145248969]\) | \(7730680381889320597382223137569/441370202660156250000\) | \(33287651039560408264160156250000\) | \([2, 2]\) | \(6688604160\) | \(5.5148\) | |
| 430950.ep3 | 430950ep7 | \([1, 1, 1, -1737279972438, -887127155104842969]\) | \(-7688701694683937879808871873249/58423707246780395507812500\) | \(-4406251186751950532197952270507812500\) | \([2]\) | \(13377208320\) | \(5.8614\) | |
| 430950.ep1 | 430950ep8 | \([1, 1, 1, -27846975025438, -56560760556113998969]\) | \(31664865542564944883878115208137569/103216295812500\) | \(7784458524600583007812500\) | \([2]\) | \(13377208320\) | \(5.8614\) |
Rank
sage: E.rank()
The elliptic curves in class 430950ep have rank \(0\).
Complex multiplication
The elliptic curves in class 430950ep do not have complex multiplication.Modular form 430950.2.a.ep
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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
