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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 13872.bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
13872.bm1 | 13872bi5 | \([0, 1, 0, -128288352, 559236806388]\) | \(2361739090258884097/5202\) | \(514308644610048\) | \([2]\) | \(884736\) | \(2.9568\) | |
13872.bm2 | 13872bi3 | \([0, 1, 0, -8018112, 8735863860]\) | \(576615941610337/27060804\) | \(2675433569261469696\) | \([2, 2]\) | \(442368\) | \(2.6103\) | |
13872.bm3 | 13872bi6 | \([0, 1, 0, -7601952, 9683543412]\) | \(-491411892194497/125563633938\) | \(-12414160396571511693312\) | \([2]\) | \(884736\) | \(2.9568\) | |
13872.bm4 | 13872bi2 | \([0, 1, 0, -527232, 121351860]\) | \(163936758817/30338064\) | \(2999448015365799936\) | \([2, 2]\) | \(221184\) | \(2.2637\) | |
13872.bm5 | 13872bi1 | \([0, 1, 0, -157312, -22325068]\) | \(4354703137/352512\) | \(34851974034751488\) | \([2]\) | \(110592\) | \(1.9171\) | \(\Gamma_0(N)\)-optimal |
13872.bm6 | 13872bi4 | \([0, 1, 0, 1044928, 708081972]\) | \(1276229915423/2927177028\) | \(-289402623953161961472\) | \([4]\) | \(442368\) | \(2.6103\) |
Rank
sage: E.rank()
The elliptic curves in class 13872.bm have rank \(1\).
Complex multiplication
The elliptic curves in class 13872.bm do not have complex multiplication.Modular form 13872.2.a.bm
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.