Show commands:
SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 37440.ef
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 37440.ef1 | 37440ex8 | \([0, 0, 0, -74880012, -249400300016]\) | \(242970740812818720001/24375\) | \(4658135040000\) | \([2]\) | \(1572864\) | \(2.7784\) | |
| 37440.ef2 | 37440ex6 | \([0, 0, 0, -4680012, -3896860016]\) | \(59319456301170001/594140625\) | \(113542041600000000\) | \([2, 2]\) | \(786432\) | \(2.4319\) | |
| 37440.ef3 | 37440ex7 | \([0, 0, 0, -4567692, -4092791024]\) | \(-55150149867714721/5950927734375\) | \(-1137240000000000000000\) | \([2]\) | \(1572864\) | \(2.7784\) | |
| 37440.ef4 | 37440ex4 | \([0, 0, 0, -299532, -57807344]\) | \(15551989015681/1445900625\) | \(276315912437760000\) | \([2, 2]\) | \(393216\) | \(2.0853\) | |
| 37440.ef5 | 37440ex2 | \([0, 0, 0, -66252, 5551504]\) | \(168288035761/27720225\) | \(5297417492889600\) | \([2, 2]\) | \(196608\) | \(1.7387\) | |
| 37440.ef6 | 37440ex1 | \([0, 0, 0, -63372, 6140176]\) | \(147281603041/5265\) | \(1006157168640\) | \([2]\) | \(98304\) | \(1.3921\) | \(\Gamma_0(N)\)-optimal |
| 37440.ef7 | 37440ex3 | \([0, 0, 0, 120948, 31235344]\) | \(1023887723039/2798036865\) | \(-534713171859210240\) | \([2]\) | \(393216\) | \(2.0853\) | |
| 37440.ef8 | 37440ex5 | \([0, 0, 0, 348468, -273720944]\) | \(24487529386319/183539412225\) | \(-35074927889488281600\) | \([2]\) | \(786432\) | \(2.4319\) |
Rank
sage: E.rank()
The elliptic curves in class 37440.ef have rank \(0\).
Complex multiplication
The elliptic curves in class 37440.ef do not have complex multiplication.Modular form 37440.2.a.ef
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 16 & 16 & 8 \\ 2 & 1 & 2 & 2 & 4 & 8 & 8 & 4 \\ 4 & 2 & 1 & 4 & 8 & 16 & 16 & 8 \\ 4 & 2 & 4 & 1 & 2 & 4 & 4 & 2 \\ 8 & 4 & 8 & 2 & 1 & 2 & 2 & 4 \\ 16 & 8 & 16 & 4 & 2 & 1 & 4 & 8 \\ 16 & 8 & 16 & 4 & 2 & 4 & 1 & 8 \\ 8 & 4 & 8 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
