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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 416955k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
416955.k4 | 416955k1 | \([1, 1, 1, -1271, 126068]\) | \(-4826809/144375\) | \(-6792249069375\) | \([2]\) | \(663552\) | \(1.1427\) | \(\Gamma_0(N)\)-optimal* |
416955.k3 | 416955k2 | \([1, 1, 1, -46396, 3808268]\) | \(234770924809/1334025\) | \(62760381401025\) | \([2, 2]\) | \(1327104\) | \(1.4892\) | \(\Gamma_0(N)\)-optimal* |
416955.k1 | 416955k3 | \([1, 1, 1, -741321, 245364198]\) | \(957681397954009/31185\) | \(1467125798985\) | \([2]\) | \(2654208\) | \(1.8358\) | \(\Gamma_0(N)\)-optimal* |
416955.k2 | 416955k4 | \([1, 1, 1, -73471, -1184362]\) | \(932288503609/527295615\) | \(24807086755111815\) | \([2]\) | \(2654208\) | \(1.8358\) |
Rank
sage: E.rank()
The elliptic curves in class 416955k have rank \(1\).
Complex multiplication
The elliptic curves in class 416955k do not have complex multiplication.Modular form 416955.2.a.k
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.