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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 2175.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2175.b1 | 2175c3 | \([1, 1, 1, -6438, -198594]\) | \(1888690601881/31827645\) | \(497306953125\) | \([2]\) | \(4608\) | \(1.0424\) | |
2175.b2 | 2175c2 | \([1, 1, 1, -813, 3906]\) | \(3803721481/1703025\) | \(26609765625\) | \([2, 2]\) | \(2304\) | \(0.69580\) | |
2175.b3 | 2175c1 | \([1, 1, 1, -688, 6656]\) | \(2305199161/1305\) | \(20390625\) | \([4]\) | \(1152\) | \(0.34922\) | \(\Gamma_0(N)\)-optimal |
2175.b4 | 2175c4 | \([1, 1, 1, 2812, 32906]\) | \(157376536199/118918125\) | \(-1858095703125\) | \([2]\) | \(4608\) | \(1.0424\) |
Rank
sage: E.rank()
The elliptic curves in class 2175.b have rank \(2\).
Complex multiplication
The elliptic curves in class 2175.b do not have complex multiplication.Modular form 2175.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.