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SageMath
E = EllipticCurve("bj1")
E.isogeny_class()
Elliptic curves in class 150075bj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
150075.bj1 | 150075bj1 | \([1, -1, 0, -84042, -9356009]\) | \(5763259856089/450225\) | \(5128344140625\) | \([2]\) | \(331776\) | \(1.4861\) | \(\Gamma_0(N)\)-optimal |
150075.bj2 | 150075bj2 | \([1, -1, 0, -78417, -10666634]\) | \(-4681768588489/1621620405\) | \(-18471269925703125\) | \([2]\) | \(663552\) | \(1.8326\) |
Rank
sage: E.rank()
The elliptic curves in class 150075bj have rank \(2\).
Complex multiplication
The elliptic curves in class 150075bj do not have complex multiplication.Modular form 150075.2.a.bj
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.