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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 23400.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 23400.bc1 | 23400bh4 | \([0, 0, 0, -547560075, -4931690062250]\) | \(1556580279686303289604/114075\) | \(1330570800000000\) | \([2]\) | \(2949120\) | \(3.2716\) | |
| 23400.bc2 | 23400bh6 | \([0, 0, 0, -120276075, 423890167750]\) | \(8248670337458940482/1446075439453125\) | \(33734047851562500000000000\) | \([2]\) | \(5898240\) | \(3.6182\) | |
| 23400.bc3 | 23400bh3 | \([0, 0, 0, -34983075, -73453315250]\) | \(405929061432816484/35083409765625\) | \(409212891506250000000000\) | \([2, 2]\) | \(2949120\) | \(3.2716\) | |
| 23400.bc4 | 23400bh2 | \([0, 0, 0, -34222575, -77057324750]\) | \(1520107298839022416/13013105625\) | \(37946216002500000000\) | \([2, 2]\) | \(1474560\) | \(2.9250\) | |
| 23400.bc5 | 23400bh1 | \([0, 0, 0, -2091450, -1260000875]\) | \(-5551350318708736/550618236675\) | \(-100350173634018750000\) | \([2]\) | \(737280\) | \(2.5784\) | \(\Gamma_0(N)\)-optimal |
| 23400.bc6 | 23400bh5 | \([0, 0, 0, 38141925, -340140190250]\) | \(263059523447441758/2294739983908125\) | \(-53531694344608740000000000\) | \([2]\) | \(5898240\) | \(3.6182\) |
Rank
sage: E.rank()
The elliptic curves in class 23400.bc have rank \(1\).
Complex multiplication
The elliptic curves in class 23400.bc do not have complex multiplication.Modular form 23400.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
