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SageMath
E = EllipticCurve("ct1")
E.isogeny_class()
Elliptic curves in class 333270.ct
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 333270.ct1 | 333270ct6 | \([1, -1, 0, -1066261371939, -423782827709553135]\) | \(1242282009445982549834550082561/41992020\) | \(4531701662460613620\) | \([2]\) | \(1660944384\) | \(4.9995\) | |
| 333270.ct2 | 333270ct4 | \([1, -1, 0, -66641335839, -6621594168308355]\) | \(303291507481995500913332161/1763329743680400\) | \(190295306844079336343312400\) | \([2, 2]\) | \(830472192\) | \(4.6530\) | |
| 333270.ct3 | 333270ct5 | \([1, -1, 0, -66602771739, -6629640498357975]\) | \(-302765284673144739899429761/731344538939408411220\) | \(-78925359221661831699703070168820\) | \([2]\) | \(1660944384\) | \(4.9995\) | |
| 333270.ct4 | 333270ct2 | \([1, -1, 0, -4167493839, -103335884164755]\) | \(74174404299602673044161/178530248806560000\) | \(19266656505597847702611360000\) | \([2, 2]\) | \(415236096\) | \(4.3064\) | |
| 333270.ct5 | 333270ct3 | \([1, -1, 0, -2634451839, -180331692181155]\) | \(-18736995756767139956161/119334500162058560400\) | \(-12878360049678656649919288592400\) | \([2]\) | \(830472192\) | \(4.6530\) | |
| 333270.ct6 | 333270ct1 | \([1, -1, 0, -358693839, -285753124755]\) | \(47293441677949844161/27041817600000000\) | \(2918303281763456025600000000\) | \([2]\) | \(207618048\) | \(3.9598\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 333270.ct have rank \(1\).
Complex multiplication
The elliptic curves in class 333270.ct do not have complex multiplication.Modular form 333270.2.a.ct
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
