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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 42a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 42.a5 | 42a1 | \([1, 1, 1, -4, 5]\) | \(-7189057/16128\) | \(-16128\) | \([8]\) | \(4\) | \(-0.50405\) | \(\Gamma_0(N)\)-optimal |
| 42.a4 | 42a2 | \([1, 1, 1, -84, 261]\) | \(65597103937/63504\) | \(63504\) | \([2, 4]\) | \(8\) | \(-0.15747\) | |
| 42.a3 | 42a3 | \([1, 1, 1, -104, 101]\) | \(124475734657/63011844\) | \(63011844\) | \([2, 2]\) | \(16\) | \(0.18910\) | |
| 42.a1 | 42a4 | \([1, 1, 1, -1344, 18405]\) | \(268498407453697/252\) | \(252\) | \([4]\) | \(16\) | \(0.18910\) | |
| 42.a2 | 42a5 | \([1, 1, 1, -914, -10915]\) | \(84448510979617/933897762\) | \(933897762\) | \([2]\) | \(32\) | \(0.53568\) | |
| 42.a6 | 42a6 | \([1, 1, 1, 386, 1277]\) | \(6359387729183/4218578658\) | \(-4218578658\) | \([2]\) | \(32\) | \(0.53568\) |
Rank
sage: E.rank()
The elliptic curves in class 42a have rank \(0\).
Complex multiplication
The elliptic curves in class 42a do not have complex multiplication.Modular form 42.2.a.a
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.
