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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 37905w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 37905.t6 | 37905w1 | \([1, 0, 1, 45262172, -35355597619]\) | \(217975805967584185919/137624180157363375\) | \(-6474650802405878614008375\) | \([2]\) | \(7050240\) | \(3.4499\) | \(\Gamma_0(N)\)-optimal |
| 37905.t5 | 37905w2 | \([1, 0, 1, -189967233, -289309263257]\) | \(16115292555782480096401/8557487595112640625\) | \(402594543058645472439515625\) | \([2, 2]\) | \(14100480\) | \(3.7964\) | |
| 37905.t3 | 37905w3 | \([1, 0, 1, -1754470838, 28068257279531]\) | \(12695229840756170655249121/112459065576416015625\) | \(5290735816479264277587890625\) | \([2, 2]\) | \(28200960\) | \(4.1430\) | |
| 37905.t2 | 37905w4 | \([1, 0, 1, -2389134108, -44898969489257]\) | \(32057060107551693105326401/40490171782737618375\) | \(1904895803360231848293663375\) | \([2]\) | \(28200960\) | \(4.1430\) | |
| 37905.t4 | 37905w5 | \([1, 0, 1, -529419143, 66539290688633]\) | \(-348819718507793207040241/40453612804412841796875\) | \(-1903175854016482830047607421875\) | \([2]\) | \(56401920\) | \(4.4896\) | |
| 37905.t1 | 37905w6 | \([1, 0, 1, -28011580213, 1804487740623281]\) | \(51667200931417724201028999121/4108467163497046875\) | \(193286457266289611132671875\) | \([4]\) | \(56401920\) | \(4.4896\) |
Rank
sage: E.rank()
The elliptic curves in class 37905w have rank \(1\).
Complex multiplication
The elliptic curves in class 37905w do not have complex multiplication.Modular form 37905.2.a.w
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
