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SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 308550.ef
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 308550.ef1 | 308550ef4 | \([1, 0, 1, -4798577226, -127943493118652]\) | \(441453577446719855661097/4354701912\) | \(120540938655072375000\) | \([2]\) | \(165150720\) | \(3.8850\) | |
| 308550.ef2 | 308550ef2 | \([1, 0, 1, -299918226, -1999035754652]\) | \(107784459654566688937/10704361149504\) | \(296303574099632121000000\) | \([2, 2]\) | \(82575360\) | \(3.5384\) | |
| 308550.ef3 | 308550ef3 | \([1, 0, 1, -277291226, -2313370038652]\) | \(-85183593440646799657/34223681512621656\) | \(-947333428815336461328375000\) | \([2]\) | \(165150720\) | \(3.8850\) | |
| 308550.ef4 | 308550ef1 | \([1, 0, 1, -20166226, -26224650652]\) | \(32765849647039657/8229948198912\) | \(227810238456449088000000\) | \([2]\) | \(41287680\) | \(3.1919\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 308550.ef have rank \(1\).
Complex multiplication
The elliptic curves in class 308550.ef do not have complex multiplication.Modular form 308550.2.a.ef
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.
