Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 3675e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3675.j7 | 3675e1 | \([1, 1, 0, -25, -8000]\) | \(-1/15\) | \(-27573984375\) | \([2]\) | \(2304\) | \(0.68225\) | \(\Gamma_0(N)\)-optimal |
3675.j6 | 3675e2 | \([1, 1, 0, -6150, -185625]\) | \(13997521/225\) | \(413609765625\) | \([2, 2]\) | \(4608\) | \(1.0288\) | |
3675.j4 | 3675e3 | \([1, 1, 0, -98025, -11853750]\) | \(56667352321/15\) | \(27573984375\) | \([2]\) | \(9216\) | \(1.3754\) | |
3675.j5 | 3675e4 | \([1, 1, 0, -12275, 237000]\) | \(111284641/50625\) | \(93062197265625\) | \([2, 2]\) | \(9216\) | \(1.3754\) | |
3675.j2 | 3675e5 | \([1, 1, 0, -165400, 25808875]\) | \(272223782641/164025\) | \(301521519140625\) | \([2, 2]\) | \(18432\) | \(1.7220\) | |
3675.j8 | 3675e6 | \([1, 1, 0, 42850, 1835625]\) | \(4733169839/3515625\) | \(-6462652587890625\) | \([2]\) | \(18432\) | \(1.7220\) | |
3675.j1 | 3675e7 | \([1, 1, 0, -2646025, 1655579500]\) | \(1114544804970241/405\) | \(744497578125\) | \([2]\) | \(36864\) | \(2.0685\) | |
3675.j3 | 3675e8 | \([1, 1, 0, -134775, 35700750]\) | \(-147281603041/215233605\) | \(-395656537416328125\) | \([2]\) | \(36864\) | \(2.0685\) |
Rank
sage: E.rank()
The elliptic curves in class 3675e have rank \(0\).
Complex multiplication
The elliptic curves in class 3675e do not have complex multiplication.Modular form 3675.2.a.e
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 2 & 2 \\ 8 & 4 & 8 & 2 & 4 & 1 & 8 & 8 \\ 16 & 8 & 16 & 4 & 2 & 8 & 1 & 4 \\ 16 & 8 & 16 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.