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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 72450.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 72450.b1 | 72450bk4 | \([1, -1, 0, -341210367, 2414269686541]\) | \(385693937170561837203625/2159357734550274048\) | \(24596434195111715328000000\) | \([2]\) | \(33177600\) | \(3.7144\) | |
| 72450.b2 | 72450bk2 | \([1, -1, 0, -25198992, -46382343584]\) | \(155355156733986861625/8291568305839392\) | \(94446145233701824500000\) | \([2]\) | \(11059200\) | \(3.1651\) | |
| 72450.b3 | 72450bk3 | \([1, -1, 0, -9434367, 79561974541]\) | \(-8152944444844179625/235342826399858688\) | \(-2680701881960890368000000\) | \([2]\) | \(16588800\) | \(3.3678\) | |
| 72450.b4 | 72450bk1 | \([1, -1, 0, 1045008, -2896035584]\) | \(11079872671250375/324440155855872\) | \(-3695576150295792000000\) | \([2]\) | \(5529600\) | \(2.8185\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 72450.b have rank \(1\).
Complex multiplication
The elliptic curves in class 72450.b do not have complex multiplication.Modular form 72450.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
