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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 12342w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12342.u4 | 12342w1 | \([1, 1, 1, -806649, -210119865]\) | \(32765849647039657/8229948198912\) | \(14579855261212741632\) | \([4]\) | \(322560\) | \(2.3871\) | \(\Gamma_0(N)\)-optimal |
| 12342.u2 | 12342w2 | \([1, 1, 1, -11996729, -15997084729]\) | \(107784459654566688937/10704361149504\) | \(18963428742376455744\) | \([2, 2]\) | \(645120\) | \(2.7337\) | |
| 12342.u1 | 12342w3 | \([1, 1, 1, -191943089, -1023624722185]\) | \(441453577446719855661097/4354701912\) | \(7714620073924632\) | \([2]\) | \(1290240\) | \(3.0803\) | |
| 12342.u3 | 12342w4 | \([1, 1, 1, -11091649, -18511396969]\) | \(-85183593440646799657/34223681512621656\) | \(-60629339444181533525016\) | \([2]\) | \(1290240\) | \(3.0803\) |
Rank
sage: E.rank()
The elliptic curves in class 12342w have rank \(0\).
Complex multiplication
The elliptic curves in class 12342w do not have complex multiplication.Modular form 12342.2.a.w
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.
