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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 15030.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
15030.f1 | 15030f3 | \([1, -1, 0, -96309, -11479887]\) | \(135518735544698449/782812500\) | \(570670312500\) | \([2]\) | \(49152\) | \(1.4463\) | |
15030.f2 | 15030f4 | \([1, -1, 0, -19629, 857385]\) | \(1147369112034769/233338896300\) | \(170104055402700\) | \([4]\) | \(49152\) | \(1.4463\) | |
15030.f3 | 15030f2 | \([1, -1, 0, -6129, -171315]\) | \(34930508298769/2510010000\) | \(1829797290000\) | \([2, 2]\) | \(24576\) | \(1.0997\) | |
15030.f4 | 15030f1 | \([1, -1, 0, 351, -11907]\) | \(6549699311/86572800\) | \(-63111571200\) | \([2]\) | \(12288\) | \(0.75313\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 15030.f have rank \(0\).
Complex multiplication
The elliptic curves in class 15030.f do not have complex multiplication.Modular form 15030.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.