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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 7056.bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7056.bd1 | 7056bp6 | \([0, 0, 0, -19266555, 32550241066]\) | \(2251439055699625/25088\) | \(8813365017182208\) | \([2]\) | \(165888\) | \(2.6285\) | |
| 7056.bd2 | 7056bp5 | \([0, 0, 0, -1203195, 509453098]\) | \(-548347731625/1835008\) | \(-644634698399612928\) | \([2]\) | \(82944\) | \(2.2819\) | |
| 7056.bd3 | 7056bp4 | \([0, 0, 0, -250635, 39587002]\) | \(4956477625/941192\) | \(330638896972726272\) | \([2]\) | \(55296\) | \(2.0792\) | |
| 7056.bd4 | 7056bp2 | \([0, 0, 0, -74235, -7779926]\) | \(128787625/98\) | \(34427207098368\) | \([2]\) | \(18432\) | \(1.5299\) | |
| 7056.bd5 | 7056bp1 | \([0, 0, 0, -3675, -173558]\) | \(-15625/28\) | \(-9836344885248\) | \([2]\) | \(9216\) | \(1.1833\) | \(\Gamma_0(N)\)-optimal |
| 7056.bd6 | 7056bp3 | \([0, 0, 0, 31605, 3629626]\) | \(9938375/21952\) | \(-7711694390034432\) | \([2]\) | \(27648\) | \(1.7326\) |
Rank
sage: E.rank()
The elliptic curves in class 7056.bd have rank \(0\).
Complex multiplication
The elliptic curves in class 7056.bd do not have complex multiplication.Modular form 7056.2.a.bd
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.
