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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 486720.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
486720.r1 | 486720r3 | \([0, 0, 0, -3416776428, 76872845487152]\) | \(19129597231400697604/26325\) | \(6070660596257587200\) | \([2]\) | \(132120576\) | \(3.7705\) | \(\Gamma_0(N)\)-optimal* |
486720.r2 | 486720r2 | \([0, 0, 0, -213550428, 1201115753552]\) | \(18681746265374416/693005625\) | \(39952535049120245760000\) | \([2, 2]\) | \(66060288\) | \(3.4239\) | \(\Gamma_0(N)\)-optimal* |
486720.r3 | 486720r4 | \([0, 0, 0, -203694348, 1316995657328]\) | \(-4053153720264484/903687890625\) | \(-208394395780904985600000000\) | \([2]\) | \(132120576\) | \(3.7705\) | |
486720.r4 | 486720r1 | \([0, 0, 0, -13964808, 16934352968]\) | \(83587439220736/13990184325\) | \(50409342780245756236800\) | \([2]\) | \(33030144\) | \(3.0773\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 486720.r have rank \(0\).
Complex multiplication
The elliptic curves in class 486720.r do not have complex multiplication.Modular form 486720.2.a.r
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.