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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 3450.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3450.o1 | 3450r4 | \([1, 1, 1, -19263, 167781]\) | \(50591419971625/28422890688\) | \(444107667000000\) | \([2]\) | \(13824\) | \(1.5007\) | |
3450.o2 | 3450r2 | \([1, 1, 1, -14388, 658281]\) | \(21081759765625/57132\) | \(892687500\) | \([2]\) | \(4608\) | \(0.95142\) | |
3450.o3 | 3450r1 | \([1, 1, 1, -888, 10281]\) | \(-4956477625/268272\) | \(-4191750000\) | \([2]\) | \(2304\) | \(0.60485\) | \(\Gamma_0(N)\)-optimal |
3450.o4 | 3450r3 | \([1, 1, 1, 4737, 23781]\) | \(752329532375/448524288\) | \(-7008192000000\) | \([2]\) | \(6912\) | \(1.1542\) |
Rank
sage: E.rank()
The elliptic curves in class 3450.o have rank \(1\).
Complex multiplication
The elliptic curves in class 3450.o do not have complex multiplication.Modular form 3450.2.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.