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SageMath
E = EllipticCurve("dr1")
E.isogeny_class()
Elliptic curves in class 84966.dr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 84966.dr1 | 84966dv6 | \([1, 0, 0, -392883079, 2997356200775]\) | \(2361739090258884097/5202\) | \(14772435969171762\) | \([2]\) | \(14155776\) | \(3.2366\) | |
| 84966.dr2 | 84966dv4 | \([1, 0, 0, -24555469, 46831048109]\) | \(576615941610337/27060804\) | \(76846211911631505924\) | \([2, 2]\) | \(7077888\) | \(2.8901\) | |
| 84966.dr3 | 84966dv5 | \([1, 0, 0, -23280979, 51909380963]\) | \(-491411892194497/125563633938\) | \(-356570692503965278126578\) | \([2]\) | \(14155776\) | \(3.2366\) | |
| 84966.dr4 | 84966dv2 | \([1, 0, 0, -1614649, 651177449]\) | \(163936758817/30338064\) | \(86152846572209715984\) | \([2, 2]\) | \(3538944\) | \(2.5435\) | |
| 84966.dr5 | 84966dv1 | \([1, 0, 0, -481769, -119407527]\) | \(4354703137/352512\) | \(1001049778616815872\) | \([2]\) | \(1769472\) | \(2.1969\) | \(\Gamma_0(N)\)-optimal |
| 84966.dr6 | 84966dv3 | \([1, 0, 0, 3200091, 3793276773]\) | \(1276229915423/2927177028\) | \(-8312482740592175684868\) | \([2]\) | \(7077888\) | \(2.8901\) |
Rank
sage: E.rank()
The elliptic curves in class 84966.dr have rank \(0\).
Complex multiplication
The elliptic curves in class 84966.dr do not have complex multiplication.Modular form 84966.2.a.dr
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
