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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 462.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
462.a1 | 462c3 | \([1, 1, 0, -226, -1406]\) | \(1285429208617/614922\) | \(614922\) | \([2]\) | \(128\) | \(0.065661\) | |
462.a2 | 462c4 | \([1, 1, 0, -126, 486]\) | \(223980311017/4278582\) | \(4278582\) | \([2]\) | \(128\) | \(0.065661\) | |
462.a3 | 462c2 | \([1, 1, 0, -16, -20]\) | \(498677257/213444\) | \(213444\) | \([2, 2]\) | \(64\) | \(-0.28091\) | |
462.a4 | 462c1 | \([1, 1, 0, 4, 0]\) | \(4657463/3696\) | \(-3696\) | \([2]\) | \(32\) | \(-0.62749\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 462.a have rank \(1\).
Complex multiplication
The elliptic curves in class 462.a do not have complex multiplication.Modular form 462.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.