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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 3450.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3450.k1 | 3450i4 | \([1, 0, 1, -6176, 186248]\) | \(1666957239793/301806\) | \(4715718750\) | \([2]\) | \(4096\) | \(0.85973\) | |
3450.k2 | 3450i3 | \([1, 0, 1, -2676, -51752]\) | \(135559106353/5037138\) | \(78705281250\) | \([2]\) | \(4096\) | \(0.85973\) | |
3450.k3 | 3450i2 | \([1, 0, 1, -426, 2248]\) | \(545338513/171396\) | \(2678062500\) | \([2, 2]\) | \(2048\) | \(0.51316\) | |
3450.k4 | 3450i1 | \([1, 0, 1, 74, 248]\) | \(2924207/3312\) | \(-51750000\) | \([2]\) | \(1024\) | \(0.16659\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3450.k have rank \(1\).
Complex multiplication
The elliptic curves in class 3450.k do not have complex multiplication.Modular form 3450.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.