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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 124950.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
124950.bc1 | 124950h4 | \([1, 1, 0, -13727375, -19581945375]\) | \(155624507032726369/175394100\) | \(322420944857812500\) | \([2]\) | \(4718592\) | \(2.6449\) | |
124950.bc2 | 124950h3 | \([1, 1, 0, -2114375, 764765625]\) | \(568671957006049/191329687500\) | \(351714787573242187500\) | \([2]\) | \(4718592\) | \(2.6449\) | |
124950.bc3 | 124950h2 | \([1, 1, 0, -864875, -301057875]\) | \(38920307374369/1274490000\) | \(2342851156406250000\) | \([2, 2]\) | \(2359296\) | \(2.2983\) | |
124950.bc4 | 124950h1 | \([1, 1, 0, 17125, -16171875]\) | \(302111711/61689600\) | \(-113401871100000000\) | \([2]\) | \(1179648\) | \(1.9517\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 124950.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 124950.bc do not have complex multiplication.Modular form 124950.2.a.bc
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.