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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 3150k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3150.f7 | 3150k1 | \([1, -1, 0, -111942, -14382284]\) | \(13619385906841/6048000\) | \(68890500000000\) | \([2]\) | \(18432\) | \(1.6138\) | \(\Gamma_0(N)\)-optimal |
| 3150.f6 | 3150k2 | \([1, -1, 0, -129942, -9432284]\) | \(21302308926361/8930250000\) | \(101721128906250000\) | \([2, 2]\) | \(36864\) | \(1.9604\) | |
| 3150.f5 | 3150k3 | \([1, -1, 0, -331317, 55868341]\) | \(353108405631241/86318776320\) | \(983224811520000000\) | \([2]\) | \(55296\) | \(2.1631\) | |
| 3150.f4 | 3150k4 | \([1, -1, 0, -980442, 367339216]\) | \(9150443179640281/184570312500\) | \(2102371215820312500\) | \([2]\) | \(73728\) | \(2.3070\) | |
| 3150.f8 | 3150k5 | \([1, -1, 0, 432558, -69619784]\) | \(785793873833639/637994920500\) | \(-7267160891320312500\) | \([2]\) | \(73728\) | \(2.3070\) | |
| 3150.f2 | 3150k6 | \([1, -1, 0, -4939317, 4226108341]\) | \(1169975873419524361/108425318400\) | \(1235032142400000000\) | \([2, 2]\) | \(110592\) | \(2.5097\) | |
| 3150.f1 | 3150k7 | \([1, -1, 0, -79027317, 270424292341]\) | \(4791901410190533590281/41160000\) | \(468838125000000\) | \([2]\) | \(221184\) | \(2.8563\) | |
| 3150.f3 | 3150k8 | \([1, -1, 0, -4579317, 4867988341]\) | \(-932348627918877961/358766164249920\) | \(-4086570839659245000000\) | \([2]\) | \(221184\) | \(2.8563\) |
Rank
sage: E.rank()
The elliptic curves in class 3150k have rank \(0\).
Complex multiplication
The elliptic curves in class 3150k do not have complex multiplication.Modular form 3150.2.a.k
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
