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SageMath
E = EllipticCurve("dm1")
E.isogeny_class()
Elliptic curves in class 117600.dm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
117600.dm1 | 117600w4 | \([0, -1, 0, -833408, 292974312]\) | \(68017239368/39375\) | \(37059435000000000\) | \([2]\) | \(1179648\) | \(2.1249\) | |
117600.dm2 | 117600w3 | \([0, -1, 0, -490408, -130116188]\) | \(13858588808/229635\) | \(216130624920000000\) | \([2]\) | \(1179648\) | \(2.1249\) | |
117600.dm3 | 117600w1 | \([0, -1, 0, -61658, 2796312]\) | \(220348864/99225\) | \(11673722025000000\) | \([2, 2]\) | \(589824\) | \(1.7783\) | \(\Gamma_0(N)\)-optimal |
117600.dm4 | 117600w2 | \([0, -1, 0, 213967, 20711937]\) | \(143877824/108045\) | \(-813528717120000000\) | \([2]\) | \(1179648\) | \(2.1249\) |
Rank
sage: E.rank()
The elliptic curves in class 117600.dm have rank \(0\).
Complex multiplication
The elliptic curves in class 117600.dm do not have complex multiplication.Modular form 117600.2.a.dm
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.