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SageMath
E = EllipticCurve("cb1")
E.isogeny_class()
Elliptic curves in class 22320cb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
22320.bm3 | 22320cb1 | \([0, 0, 0, -15627, -668486]\) | \(141339344329/17141760\) | \(51185021091840\) | \([2]\) | \(55296\) | \(1.3610\) | \(\Gamma_0(N)\)-optimal |
22320.bm2 | 22320cb2 | \([0, 0, 0, -61707, 5202106]\) | \(8702409880009/1120910400\) | \(3347020519833600\) | \([2, 2]\) | \(110592\) | \(1.7076\) | |
22320.bm4 | 22320cb3 | \([0, 0, 0, 93813, 27192634]\) | \(30579142915511/124675335000\) | \(-372278555504640000\) | \([4]\) | \(221184\) | \(2.0542\) | |
22320.bm1 | 22320cb4 | \([0, 0, 0, -954507, 358929466]\) | \(32208729120020809/658986840\) | \(1967724160450560\) | \([2]\) | \(221184\) | \(2.0542\) |
Rank
sage: E.rank()
The elliptic curves in class 22320cb have rank \(1\).
Complex multiplication
The elliptic curves in class 22320cb do not have complex multiplication.Modular form 22320.2.a.cb
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.