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SageMath
E = EllipticCurve("qf1")
E.isogeny_class()
Elliptic curves in class 369600qf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
369600.qf6 | 369600qf1 | \([0, 1, 0, -458453633, 3778101904863]\) | \(2601656892010848045529/56330588160\) | \(230730089103360000000\) | \([2]\) | \(63700992\) | \(3.4342\) | \(\Gamma_0(N)\)-optimal |
369600.qf5 | 369600qf2 | \([0, 1, 0, -458965633, 3769239696863]\) | \(2610383204210122997209/12104550027662400\) | \(49580236913305190400000000\) | \([2, 2]\) | \(127401984\) | \(3.7808\) | |
369600.qf4 | 369600qf3 | \([0, 1, 0, -489197633, 3242454232863]\) | \(3160944030998056790089/720291785342976000\) | \(2950315152764829696000000000\) | \([2]\) | \(191102976\) | \(3.9835\) | |
369600.qf7 | 369600qf4 | \([0, 1, 0, -225685633, 7598064336863]\) | \(-310366976336070130009/5909282337130963560\) | \(-24204420452888426741760000000\) | \([2]\) | \(254803968\) | \(4.1274\) | |
369600.qf3 | 369600qf5 | \([0, 1, 0, -700437633, -626758063137]\) | \(9278380528613437145689/5328033205714065000\) | \(21823624010604810240000000000\) | \([2]\) | \(254803968\) | \(4.1274\) | |
369600.qf2 | 369600qf6 | \([0, 1, 0, -2586349633, -47854654247137]\) | \(467116778179943012100169/28800309694464000000\) | \(117966068508524544000000000000\) | \([2, 2]\) | \(382205952\) | \(4.3301\) | |
369600.qf8 | 369600qf7 | \([0, 1, 0, 2021650367, -199821886247137]\) | \(223090928422700449019831/4340371122724101696000\) | \(-17778160118677920546816000000000\) | \([2]\) | \(764411904\) | \(4.6767\) | |
369600.qf1 | 369600qf8 | \([0, 1, 0, -40748781633, -3166068810535137]\) | \(1826870018430810435423307849/7641104625000000000\) | \(31297964544000000000000000000\) | \([2]\) | \(764411904\) | \(4.6767\) |
Rank
sage: E.rank()
The elliptic curves in class 369600qf have rank \(1\).
Complex multiplication
The elliptic curves in class 369600qf do not have complex multiplication.Modular form 369600.2.a.qf
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.