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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 6930.o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6930.o1 | 6930p4 | \([1, -1, 0, -235044, 42676308]\) | \(1969902499564819009/63690429687500\) | \(46430323242187500\) | \([6]\) | \(82944\) | \(1.9720\) | |
| 6930.o2 | 6930p2 | \([1, -1, 0, -32184, -2194560]\) | \(5057359576472449/51765560000\) | \(37737093240000\) | \([2]\) | \(27648\) | \(1.4227\) | |
| 6930.o3 | 6930p1 | \([1, -1, 0, -504, -84672]\) | \(-19443408769/4249907200\) | \(-3098182348800\) | \([2]\) | \(13824\) | \(1.0761\) | \(\Gamma_0(N)\)-optimal |
| 6930.o4 | 6930p3 | \([1, -1, 0, 4536, 2283120]\) | \(14156681599871/3100231750000\) | \(-2260068945750000\) | \([6]\) | \(41472\) | \(1.6254\) |
Rank
sage: E.rank()
The elliptic curves in class 6930.o have rank \(1\).
Complex multiplication
The elliptic curves in class 6930.o do not have complex multiplication.Modular form 6930.2.a.o
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
