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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 177870bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 177870.jc3 | 177870bc1 | \([1, 0, 0, -1176423410, -15492224761980]\) | \(863913648706111516969/2486234429521920\) | \(518186897260175692797050880\) | \([2]\) | \(144506880\) | \(3.9971\) | \(\Gamma_0(N)\)-optimal |
| 177870.jc2 | 177870bc2 | \([1, 0, 0, -1662127090, -1481907830908]\) | \(2436531580079063806249/1405478914998681600\) | \(292933260628930539369090662400\) | \([2, 2]\) | \(289013760\) | \(4.3437\) | |
| 177870.jc1 | 177870bc3 | \([1, 0, 0, -17732088690, 905542078788612]\) | \(2958414657792917260183849/12401051653985258880\) | \(2584656701330237737890422440320\) | \([2]\) | \(578027520\) | \(4.6903\) | |
| 177870.jc4 | 177870bc4 | \([1, 0, 0, 6636575630, -11843668047100]\) | \(155099895405729262880471/90047655797243760000\) | \(-18767946742696583265303494640000\) | \([2]\) | \(578027520\) | \(4.6903\) |
Rank
sage: E.rank()
The elliptic curves in class 177870bc have rank \(0\).
Complex multiplication
The elliptic curves in class 177870bc do not have complex multiplication.Modular form 177870.2.a.bc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
