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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 45486.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
45486.m1 | 45486p6 | \([1, -1, 0, -8871462, -10168249676]\) | \(2251439055699625/25088\) | \(860429268582912\) | \([2]\) | \(1026432\) | \(2.4346\) | |
45486.m2 | 45486p5 | \([1, -1, 0, -554022, -159042380]\) | \(-548347731625/1835008\) | \(-62934255073492992\) | \([2]\) | \(513216\) | \(2.0881\) | |
45486.m3 | 45486p4 | \([1, -1, 0, -115407, -12340283]\) | \(4956477625/941192\) | \(32279541779180808\) | \([2]\) | \(342144\) | \(1.8853\) | |
45486.m4 | 45486p2 | \([1, -1, 0, -34182, 2439418]\) | \(128787625/98\) | \(3361051830402\) | \([2]\) | \(114048\) | \(1.3360\) | |
45486.m5 | 45486p1 | \([1, -1, 0, -1692, 54652]\) | \(-15625/28\) | \(-960300522972\) | \([2]\) | \(57024\) | \(0.98944\) | \(\Gamma_0(N)\)-optimal |
45486.m6 | 45486p3 | \([1, -1, 0, 14553, -1137731]\) | \(9938375/21952\) | \(-752875610010048\) | \([2]\) | \(171072\) | \(1.5387\) |
Rank
sage: E.rank()
The elliptic curves in class 45486.m have rank \(0\).
Complex multiplication
The elliptic curves in class 45486.m do not have complex multiplication.Modular form 45486.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.