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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 240.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
240.b1 | 240b8 | \([0, -1, 0, -85336, 9623536]\) | \(16778985534208729/81000\) | \(331776000\) | \([2]\) | \(576\) | \(1.2573\) | |
240.b2 | 240b7 | \([0, -1, 0, -7256, 34800]\) | \(10316097499609/5859375000\) | \(24000000000000\) | \([2]\) | \(576\) | \(1.2573\) | |
240.b3 | 240b6 | \([0, -1, 0, -5336, 151536]\) | \(4102915888729/9000000\) | \(36864000000\) | \([2, 2]\) | \(288\) | \(0.91074\) | |
240.b4 | 240b4 | \([0, -1, 0, -4616, -119184]\) | \(2656166199049/33750\) | \(138240000\) | \([2]\) | \(192\) | \(0.70801\) | |
240.b5 | 240b5 | \([0, -1, 0, -1096, 12400]\) | \(35578826569/5314410\) | \(21767823360\) | \([2]\) | \(192\) | \(0.70801\) | |
240.b6 | 240b2 | \([0, -1, 0, -296, -1680]\) | \(702595369/72900\) | \(298598400\) | \([2, 2]\) | \(96\) | \(0.36143\) | |
240.b7 | 240b3 | \([0, -1, 0, -216, 4080]\) | \(-273359449/1536000\) | \(-6291456000\) | \([2]\) | \(144\) | \(0.56417\) | |
240.b8 | 240b1 | \([0, -1, 0, 24, -144]\) | \(357911/2160\) | \(-8847360\) | \([2]\) | \(48\) | \(0.014860\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 240.b have rank \(0\).
Complex multiplication
The elliptic curves in class 240.b do not have complex multiplication.Modular form 240.2.a.b
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.