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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 240.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
240.d1 | 240d7 | \([0, 1, 0, -34560, 2461428]\) | \(1114544804970241/405\) | \(1658880\) | \([4]\) | \(256\) | \(0.98402\) | |
240.d2 | 240d5 | \([0, 1, 0, -2160, 37908]\) | \(272223782641/164025\) | \(671846400\) | \([2, 4]\) | \(128\) | \(0.63744\) | |
240.d3 | 240d8 | \([0, 1, 0, -1760, 52788]\) | \(-147281603041/215233605\) | \(-881596846080\) | \([4]\) | \(256\) | \(0.98402\) | |
240.d4 | 240d3 | \([0, 1, 0, -1280, -18060]\) | \(56667352321/15\) | \(61440\) | \([2]\) | \(64\) | \(0.29087\) | |
240.d5 | 240d4 | \([0, 1, 0, -160, 308]\) | \(111284641/50625\) | \(207360000\) | \([2, 4]\) | \(64\) | \(0.29087\) | |
240.d6 | 240d2 | \([0, 1, 0, -80, -300]\) | \(13997521/225\) | \(921600\) | \([2, 2]\) | \(32\) | \(-0.055704\) | |
240.d7 | 240d1 | \([0, 1, 0, 0, -12]\) | \(-1/15\) | \(-61440\) | \([2]\) | \(16\) | \(-0.40228\) | \(\Gamma_0(N)\)-optimal |
240.d8 | 240d6 | \([0, 1, 0, 560, 2900]\) | \(4733169839/3515625\) | \(-14400000000\) | \([4]\) | \(128\) | \(0.63744\) |
Rank
sage: E.rank()
The elliptic curves in class 240.d have rank \(0\).
Complex multiplication
The elliptic curves in class 240.d do not have complex multiplication.Modular form 240.2.a.d
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.