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SageMath
E = EllipticCurve("ea1")
E.isogeny_class()
Elliptic curves in class 333270.ea
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
333270.ea1 | 333270ea3 | \([1, -1, 1, -598188803, 3157747962027]\) | \(219353215817909485369/87028564162480920\) | \(9391962779991888106110914520\) | \([2]\) | \(311427072\) | \(4.0658\) | |
333270.ea2 | 333270ea2 | \([1, -1, 1, -271584203, -1687888525413]\) | \(20527812941011798969/474091398849600\) | \(51163072896350548591617600\) | \([2, 2]\) | \(155713536\) | \(3.7192\) | |
333270.ea3 | 333270ea1 | \([1, -1, 1, -270060683, -1708137325029]\) | \(20184279492242626489/11148103680\) | \(1203082870982136238080\) | \([2]\) | \(77856768\) | \(3.3727\) | \(\Gamma_0(N)\)-optimal |
333270.ea4 | 333270ea4 | \([1, -1, 1, 30644077, -5237620119669]\) | \(29489595518609351/109830613939935000\) | \(-11852718106456257268539735000\) | \([2]\) | \(311427072\) | \(4.0658\) |
Rank
sage: E.rank()
The elliptic curves in class 333270.ea have rank \(1\).
Complex multiplication
The elliptic curves in class 333270.ea do not have complex multiplication.Modular form 333270.2.a.ea
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.