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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 50400.ca
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
50400.ca1 | 50400df4 | \([0, 0, 0, -22963500075, -1339384407812750]\) | \(229625675762164624948320008/9568125\) | \(55801305000000000\) | \([2]\) | \(41287680\) | \(4.1140\) | |
50400.ca2 | 50400df3 | \([0, 0, 0, -1440140700, -20777112344000]\) | \(7079962908642659949376/100085966990454375\) | \(4669610875906639320000000000\) | \([2]\) | \(41287680\) | \(4.1140\) | |
50400.ca3 | 50400df1 | \([0, 0, 0, -1435218825, -20927879219000]\) | \(448487713888272974160064/91549016015625\) | \(66739232675390625000000\) | \([2, 2]\) | \(20643840\) | \(3.7674\) | \(\Gamma_0(N)\)-optimal |
50400.ca4 | 50400df2 | \([0, 0, 0, -1430298075, -21078508297250]\) | \(-55486311952875723077768/801237030029296875\) | \(-4672814359130859375000000000\) | \([2]\) | \(41287680\) | \(4.1140\) |
Rank
sage: E.rank()
The elliptic curves in class 50400.ca have rank \(1\).
Complex multiplication
The elliptic curves in class 50400.ca do not have complex multiplication.Modular form 50400.2.a.ca
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.