Show commands:
SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 35322.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
35322.i1 | 35322c4 | \([1, 1, 0, -24748124, 47372375328]\) | \(2818140246756887473/314406208368\) | \(187016145004471750128\) | \([2]\) | \(3870720\) | \(2.9184\) | |
35322.i2 | 35322c2 | \([1, 1, 0, -1671084, 613676880]\) | \(867622835347633/227964231936\) | \(135598441509385779456\) | \([2, 2]\) | \(1935360\) | \(2.5718\) | |
35322.i3 | 35322c1 | \([1, 1, 0, -594604, -168924080]\) | \(39085920587953/1955659776\) | \(1163272042706436096\) | \([2]\) | \(967680\) | \(2.2252\) | \(\Gamma_0(N)\)-optimal |
35322.i4 | 35322c3 | \([1, 1, 0, 4182276, 3962969472]\) | \(13601087408654927/19267071783792\) | \(-11460503624380551413232\) | \([2]\) | \(3870720\) | \(2.9184\) |
Rank
sage: E.rank()
The elliptic curves in class 35322.i have rank \(1\).
Complex multiplication
The elliptic curves in class 35322.i do not have complex multiplication.Modular form 35322.2.a.i
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.