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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 84150bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 84150.bh4 | 84150bc1 | \([1, -1, 0, -1499967, -531571059]\) | \(32765849647039657/8229948198912\) | \(93744253703232000000\) | \([2]\) | \(2752512\) | \(2.5422\) | \(\Gamma_0(N)\)-optimal |
| 84150.bh2 | 84150bc2 | \([1, -1, 0, -22307967, -40545355059]\) | \(107784459654566688937/10704361149504\) | \(121929363718569000000\) | \([2, 2]\) | \(5505024\) | \(2.8888\) | |
| 84150.bh3 | 84150bc3 | \([1, -1, 0, -20624967, -46922242059]\) | \(-85183593440646799657/34223681512621656\) | \(-389829122229706050375000\) | \([2]\) | \(11010048\) | \(3.2354\) | |
| 84150.bh1 | 84150bc4 | \([1, -1, 0, -356918967, -2595300340059]\) | \(441453577446719855661097/4354701912\) | \(49602776466375000\) | \([2]\) | \(11010048\) | \(3.2354\) |
Rank
sage: E.rank()
The elliptic curves in class 84150bc have rank \(0\).
Complex multiplication
The elliptic curves in class 84150bc do not have complex multiplication.Modular form 84150.2.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.
