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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 378.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 378.b1 | 378f1 | \([1, -1, 0, -141, 681]\) | \(-11527859979/28\) | \(-756\) | \([3]\) | \(72\) | \(-0.20736\) | \(\Gamma_0(N)\)-optimal |
| 378.b2 | 378f2 | \([1, -1, 0, -96, 1088]\) | \(-5000211/21952\) | \(-432081216\) | \([3]\) | \(216\) | \(0.34195\) | |
| 378.b3 | 378f3 | \([1, -1, 0, 849, -25939]\) | \(381790581/1835008\) | \(-325066162176\) | \([]\) | \(648\) | \(0.89125\) |
Rank
sage: E.rank()
The elliptic curves in class 378.b have rank \(1\).
Complex multiplication
The elliptic curves in class 378.b do not have complex multiplication.Modular form 378.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
