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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 40560bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 40560.h6 | 40560bg1 | \([0, -1, 0, -297496, -62354384]\) | \(147281603041/5265\) | \(104092259880960\) | \([2]\) | \(258048\) | \(1.7787\) | \(\Gamma_0(N)\)-optimal |
| 40560.h5 | 40560bg2 | \([0, -1, 0, -311016, -56362320]\) | \(168288035761/27720225\) | \(548045748273254400\) | \([2, 2]\) | \(516096\) | \(2.1253\) | |
| 40560.h7 | 40560bg3 | \([0, -1, 0, 567784, -317893200]\) | \(1023887723039/2798036865\) | \(-55318894683397263360\) | \([2]\) | \(1032192\) | \(2.4719\) | |
| 40560.h4 | 40560bg4 | \([0, -1, 0, -1406136, 588444336]\) | \(15551989015681/1445900625\) | \(28586336869808640000\) | \([2, 2]\) | \(1032192\) | \(2.4719\) | |
| 40560.h8 | 40560bg5 | \([0, -1, 0, 1635864, 2783551536]\) | \(24487529386319/183539412225\) | \(-3628686077060464742400\) | \([2]\) | \(2064384\) | \(2.8185\) | |
| 40560.h2 | 40560bg6 | \([0, -1, 0, -21970056, 39643441200]\) | \(59319456301170001/594140625\) | \(11746522382400000000\) | \([2, 2]\) | \(2064384\) | \(2.8185\) | |
| 40560.h3 | 40560bg7 | \([0, -1, 0, -21442776, 41636137776]\) | \(-55150149867714721/5950927734375\) | \(-117653469375000000000000\) | \([2]\) | \(4128768\) | \(3.1650\) | |
| 40560.h1 | 40560bg8 | \([0, -1, 0, -351520056, 2536841521200]\) | \(242970740812818720001/24375\) | \(481908610560000\) | \([2]\) | \(4128768\) | \(3.1650\) |
Rank
sage: E.rank()
The elliptic curves in class 40560bg have rank \(0\).
Complex multiplication
The elliptic curves in class 40560bg do not have complex multiplication.Modular form 40560.2.a.bg
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
