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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 55488.cr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55488.cr1 | 55488bl6 | \([0, 1, 0, -513153409, -4474407604513]\) | \(2361739090258884097/5202\) | \(32915753255043072\) | \([2]\) | \(7077888\) | \(3.3034\) | |
| 55488.cr2 | 55488bl4 | \([0, 1, 0, -32072449, -69918983329]\) | \(576615941610337/27060804\) | \(171227748432734060544\) | \([2, 2]\) | \(3538944\) | \(2.9568\) | |
| 55488.cr3 | 55488bl5 | \([0, 1, 0, -30407809, -77498755105]\) | \(-491411892194497/125563633938\) | \(-794506265380576748371968\) | \([2]\) | \(7077888\) | \(3.3034\) | |
| 55488.cr4 | 55488bl2 | \([0, 1, 0, -2108929, -972923809]\) | \(163936758817/30338064\) | \(191964672983411195904\) | \([2, 2]\) | \(1769472\) | \(2.6103\) | |
| 55488.cr5 | 55488bl1 | \([0, 1, 0, -629249, 177971295]\) | \(4354703137/352512\) | \(2230526338224095232\) | \([2]\) | \(884736\) | \(2.2637\) | \(\Gamma_0(N)\)-optimal |
| 55488.cr6 | 55488bl3 | \([0, 1, 0, 4179711, -5660476065]\) | \(1276229915423/2927177028\) | \(-18521767933002365534208\) | \([2]\) | \(3538944\) | \(2.9568\) |
Rank
sage: E.rank()
The elliptic curves in class 55488.cr have rank \(2\).
Complex multiplication
The elliptic curves in class 55488.cr do not have complex multiplication.Modular form 55488.2.a.cr
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.
