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SageMath
E = EllipticCurve("hf1")
E.isogeny_class()
Elliptic curves in class 141120hf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 141120.dp4 | 141120hf1 | \([0, 0, 0, -68208, 6851768]\) | \(10788913152/8575\) | \(27892413158400\) | \([2]\) | \(442368\) | \(1.5103\) | \(\Gamma_0(N)\)-optimal |
| 141120.dp3 | 141120hf2 | \([0, 0, 0, -82908, 3682448]\) | \(1210991472/588245\) | \(30614712682659840\) | \([2]\) | \(884736\) | \(1.8568\) | |
| 141120.dp2 | 141120hf3 | \([0, 0, 0, -232848, -35636328]\) | \(588791808/109375\) | \(259356749904000000\) | \([2]\) | \(1327104\) | \(2.0596\) | |
| 141120.dp1 | 141120hf4 | \([0, 0, 0, -3540348, -2563889328]\) | \(129348709488/6125\) | \(232383647913984000\) | \([2]\) | \(2654208\) | \(2.4062\) |
Rank
sage: E.rank()
The elliptic curves in class 141120hf have rank \(1\).
Complex multiplication
The elliptic curves in class 141120hf do not have complex multiplication.Modular form 141120.2.a.hf
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
