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SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 6720.bb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6720.bb1 | 6720l7 | \([0, -1, 0, -412865, 64377537]\) | \(29689921233686449/10380965400750\) | \(2721307794014208000\) | \([4]\) | \(110592\) | \(2.2387\) | |
6720.bb2 | 6720l4 | \([0, -1, 0, -368705, 86295105]\) | \(21145699168383889/2593080\) | \(679760363520\) | \([4]\) | \(36864\) | \(1.6894\) | |
6720.bb3 | 6720l6 | \([0, -1, 0, -172865, -26870463]\) | \(2179252305146449/66177562500\) | \(17348050944000000\) | \([2, 2]\) | \(55296\) | \(1.8922\) | |
6720.bb4 | 6720l3 | \([0, -1, 0, -171585, -27299775]\) | \(2131200347946769/2058000\) | \(539492352000\) | \([2]\) | \(27648\) | \(1.5456\) | |
6720.bb5 | 6720l2 | \([0, -1, 0, -23105, 1346625]\) | \(5203798902289/57153600\) | \(14982473318400\) | \([2, 2]\) | \(18432\) | \(1.3429\) | |
6720.bb6 | 6720l5 | \([0, -1, 0, -5185, 3364417]\) | \(-58818484369/18600435000\) | \(-4875992432640000\) | \([2]\) | \(36864\) | \(1.6894\) | |
6720.bb7 | 6720l1 | \([0, -1, 0, -2625, -17343]\) | \(7633736209/3870720\) | \(1014686023680\) | \([2]\) | \(9216\) | \(0.99629\) | \(\Gamma_0(N)\)-optimal |
6720.bb8 | 6720l8 | \([0, -1, 0, 46655, -90662975]\) | \(42841933504271/13565917968750\) | \(-3556224000000000000\) | \([2]\) | \(110592\) | \(2.2387\) |
Rank
sage: E.rank()
The elliptic curves in class 6720.bb have rank \(1\).
Complex multiplication
The elliptic curves in class 6720.bb do not have complex multiplication.Modular form 6720.2.a.bb
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.