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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 198744x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
198744.di4 | 198744x1 | \([0, 1, 0, -60727, -37391782]\) | \(-2725888/64827\) | \(-589012160032990512\) | \([2]\) | \(2211840\) | \(2.0905\) | \(\Gamma_0(N)\)-optimal |
198744.di3 | 198744x2 | \([0, 1, 0, -2089572, -1158125760]\) | \(6940769488/35721\) | \(5192923533352079616\) | \([2, 2]\) | \(4423680\) | \(2.4370\) | |
198744.di2 | 198744x3 | \([0, 1, 0, -3248912, 269253648]\) | \(6522128932/3720087\) | \(2163223574750666308608\) | \([2]\) | \(8847360\) | \(2.7836\) | |
198744.di1 | 198744x4 | \([0, 1, 0, -33391752, -74280018240]\) | \(7080974546692/189\) | \(109903143563006976\) | \([2]\) | \(8847360\) | \(2.7836\) |
Rank
sage: E.rank()
The elliptic curves in class 198744x have rank \(0\).
Complex multiplication
The elliptic curves in class 198744x do not have complex multiplication.Modular form 198744.2.a.x
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.