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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 10005e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
10005.d4 | 10005e1 | \([1, 1, 1, -5250, -144690]\) | \(16003198512756001/488525390625\) | \(488525390625\) | \([4]\) | \(12288\) | \(1.0189\) | \(\Gamma_0(N)\)-optimal |
10005.d2 | 10005e2 | \([1, 1, 1, -83375, -9300940]\) | \(64096096056024006001/62562515625\) | \(62562515625\) | \([2, 2]\) | \(24576\) | \(1.3654\) | |
10005.d1 | 10005e3 | \([1, 1, 1, -1334000, -593592940]\) | \(262537424941059264096001/250125\) | \(250125\) | \([2]\) | \(49152\) | \(1.7120\) | |
10005.d3 | 10005e4 | \([1, 1, 1, -82750, -9446440]\) | \(-62665433378363916001/2004003001000125\) | \(-2004003001000125\) | \([2]\) | \(49152\) | \(1.7120\) |
Rank
sage: E.rank()
The elliptic curves in class 10005e have rank \(1\).
Complex multiplication
The elliptic curves in class 10005e do not have complex multiplication.Modular form 10005.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.