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SageMath
E = EllipticCurve("bh1")
E.isogeny_class()
Elliptic curves in class 45486bh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 45486.bl5 | 45486bh1 | \([1, -1, 1, -13064, 1272395]\) | \(-7189057/16128\) | \(-553133101231872\) | \([2]\) | \(221184\) | \(1.5175\) | \(\Gamma_0(N)\)-optimal |
| 45486.bl4 | 45486bh2 | \([1, -1, 1, -272984, 54919883]\) | \(65597103937/63504\) | \(2177961586100496\) | \([2, 2]\) | \(442368\) | \(1.8641\) | |
| 45486.bl3 | 45486bh3 | \([1, -1, 1, -337964, 26848523]\) | \(124475734657/63011844\) | \(2161082383808217156\) | \([2, 2]\) | \(884736\) | \(2.2106\) | |
| 45486.bl1 | 45486bh4 | \([1, -1, 1, -4366724, 3513311435]\) | \(268498407453697/252\) | \(8642704706748\) | \([2]\) | \(884736\) | \(2.2106\) | |
| 45486.bl6 | 45486bh5 | \([1, -1, 1, 1254046, 206427251]\) | \(6359387729183/4218578658\) | \(-144682260409854211842\) | \([2]\) | \(1769472\) | \(2.5572\) | |
| 45486.bl2 | 45486bh6 | \([1, -1, 1, -2969654, -1950077005]\) | \(84448510979617/933897762\) | \(32029375330392156738\) | \([2]\) | \(1769472\) | \(2.5572\) |
Rank
sage: E.rank()
The elliptic curves in class 45486bh have rank \(0\).
Complex multiplication
The elliptic curves in class 45486bh do not have complex multiplication.Modular form 45486.2.a.bh
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
