Show commands:
SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 64400.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
64400.i1 | 64400bt3 | \([0, 1, 0, -676408, 212047188]\) | \(534774372149809/5323062500\) | \(340676000000000000\) | \([2]\) | \(995328\) | \(2.1828\) | |
64400.i2 | 64400bt4 | \([0, 1, 0, -176408, 519047188]\) | \(-9486391169809/1813439640250\) | \(-116060136976000000000\) | \([2]\) | \(1990656\) | \(2.5294\) | |
64400.i3 | 64400bt1 | \([0, 1, 0, -60408, -5568812]\) | \(380920459249/12622400\) | \(807833600000000\) | \([2]\) | \(331776\) | \(1.6335\) | \(\Gamma_0(N)\)-optimal |
64400.i4 | 64400bt2 | \([0, 1, 0, 19592, -19168812]\) | \(12994449551/2489452840\) | \(-159324981760000000\) | \([2]\) | \(663552\) | \(1.9801\) |
Rank
sage: E.rank()
The elliptic curves in class 64400.i have rank \(1\).
Complex multiplication
The elliptic curves in class 64400.i do not have complex multiplication.Modular form 64400.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.