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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 27930g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
27930.g3 | 27930g1 | \([1, 1, 0, -645698, 199409652]\) | \(253060782505556761/41184460800\) | \(4845310628659200\) | \([2]\) | \(294912\) | \(2.0187\) | \(\Gamma_0(N)\)-optimal |
27930.g2 | 27930g2 | \([1, 1, 0, -708418, 158252788]\) | \(334199035754662681/101099003040000\) | \(11894196608652960000\) | \([2, 2]\) | \(589824\) | \(2.3653\) | |
27930.g4 | 27930g3 | \([1, 1, 0, 1937582, 1064772388]\) | \(6837784281928633319/8113766016106800\) | \(-954576458028948913200\) | \([2]\) | \(1179648\) | \(2.7119\) | |
27930.g1 | 27930g4 | \([1, 1, 0, -4357938, -3381051708]\) | \(77799851782095807001/3092322318750000\) | \(363808628478618750000\) | \([2]\) | \(1179648\) | \(2.7119\) |
Rank
sage: E.rank()
The elliptic curves in class 27930g have rank \(1\).
Complex multiplication
The elliptic curves in class 27930g do not have complex multiplication.Modular form 27930.2.a.g
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 Cremona 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 Cremona labels.