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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 56925m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
56925.l6 | 56925m1 | \([1, -1, 1, 6970, 654972]\) | \(3288008303/18259263\) | \(-207984417609375\) | \([2]\) | \(131072\) | \(1.4294\) | \(\Gamma_0(N)\)-optimal |
56925.l5 | 56925m2 | \([1, -1, 1, -84155, 8491722]\) | \(5786435182177/627352209\) | \(7145933755640625\) | \([2, 2]\) | \(262144\) | \(1.7760\) | |
56925.l4 | 56925m3 | \([1, -1, 1, -317030, -59507778]\) | \(309368403125137/44372288367\) | \(505428097180359375\) | \([2]\) | \(524288\) | \(2.1226\) | |
56925.l2 | 56925m4 | \([1, -1, 1, -1309280, 576949722]\) | \(21790813729717297/304746849\) | \(3471257076890625\) | \([2, 2]\) | \(524288\) | \(2.1226\) | |
56925.l3 | 56925m5 | \([1, -1, 1, -1272155, 611178972]\) | \(-19989223566735457/2584262514273\) | \(-29436365201640890625\) | \([2]\) | \(1048576\) | \(2.4691\) | |
56925.l1 | 56925m6 | \([1, -1, 1, -20948405, 36909330972]\) | \(89254274298475942657/17457\) | \(198846140625\) | \([2]\) | \(1048576\) | \(2.4691\) |
Rank
sage: E.rank()
The elliptic curves in class 56925m have rank \(0\).
Complex multiplication
The elliptic curves in class 56925m do not have complex multiplication.Modular form 56925.2.a.m
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.