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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 3990.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3990.p1 | 3990p3 | \([1, 0, 1, -88938, 9844588]\) | \(77799851782095807001/3092322318750000\) | \(3092322318750000\) | \([4]\) | \(24576\) | \(1.7389\) | |
| 3990.p2 | 3990p2 | \([1, 0, 1, -14458, -463444]\) | \(334199035754662681/101099003040000\) | \(101099003040000\) | \([2, 2]\) | \(12288\) | \(1.3924\) | |
| 3990.p3 | 3990p1 | \([1, 0, 1, -13178, -583252]\) | \(253060782505556761/41184460800\) | \(41184460800\) | \([2]\) | \(6144\) | \(1.0458\) | \(\Gamma_0(N)\)-optimal |
| 3990.p4 | 3990p4 | \([1, 0, 1, 39542, -3098644]\) | \(6837784281928633319/8113766016106800\) | \(-8113766016106800\) | \([2]\) | \(24576\) | \(1.7389\) |
Rank
sage: E.rank()
The elliptic curves in class 3990.p have rank \(1\).
Complex multiplication
The elliptic curves in class 3990.p do not have complex multiplication.Modular form 3990.2.a.p
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.
