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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 12870.m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12870.m1 | 12870t3 | \([1, -1, 0, -54954, -4944740]\) | \(25176685646263969/57915000\) | \(42220035000\) | \([2]\) | \(36864\) | \(1.2823\) | |
| 12870.m2 | 12870t2 | \([1, -1, 0, -3474, -74732]\) | \(6361447449889/294465600\) | \(214665422400\) | \([2, 2]\) | \(18432\) | \(0.93574\) | |
| 12870.m3 | 12870t1 | \([1, -1, 0, -594, 4180]\) | \(31824875809/8785920\) | \(6404935680\) | \([2]\) | \(9216\) | \(0.58916\) | \(\Gamma_0(N)\)-optimal |
| 12870.m4 | 12870t4 | \([1, -1, 0, 1926, -289652]\) | \(1083523132511/50179392120\) | \(-36580776855480\) | \([2]\) | \(36864\) | \(1.2823\) |
Rank
sage: E.rank()
The elliptic curves in class 12870.m have rank \(1\).
Complex multiplication
The elliptic curves in class 12870.m do not have complex multiplication.Modular form 12870.2.a.m
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.
