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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 6930.bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6930.bd1 | 6930bg3 | \([1, -1, 1, -1368707, -615322461]\) | \(388980071198593573609/486165942108000\) | \(354414971796732000\) | \([2]\) | \(122880\) | \(2.2764\) | |
| 6930.bd2 | 6930bg2 | \([1, -1, 1, -108707, -3970461]\) | \(194878967635813609/103306896000000\) | \(75310727184000000\) | \([2, 2]\) | \(61440\) | \(1.9298\) | |
| 6930.bd3 | 6930bg1 | \([1, -1, 1, -62627, 6001251]\) | \(37262716093162729/333053952000\) | \(242796331008000\) | \([4]\) | \(30720\) | \(1.5833\) | \(\Gamma_0(N)\)-optimal |
| 6930.bd4 | 6930bg4 | \([1, -1, 1, 414013, -31360989]\) | \(10765621376623941911/6809085937500000\) | \(-4963823648437500000\) | \([2]\) | \(122880\) | \(2.2764\) |
Rank
sage: E.rank()
The elliptic curves in class 6930.bd have rank \(1\).
Complex multiplication
The elliptic curves in class 6930.bd do not have complex multiplication.Modular form 6930.2.a.bd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
