Show commands:
SageMath
E = EllipticCurve("bh1")
E.isogeny_class()
Elliptic curves in class 212940bh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
212940.i3 | 212940bh1 | \([0, 0, 0, -1918488, -1671870863]\) | \(-13870539341824/13420809675\) | \(-755590244983595002800\) | \([2]\) | \(6967296\) | \(2.7023\) | \(\Gamma_0(N)\)-optimal |
212940.i2 | 212940bh2 | \([0, 0, 0, -35829183, -82521749882]\) | \(5646857395652944/2031631875\) | \(1830090748110075360000\) | \([2]\) | \(13934592\) | \(3.0488\) | |
212940.i4 | 212940bh3 | \([0, 0, 0, 15968472, 29010525973]\) | \(7998456195055616/11086576921875\) | \(-624173173995107850750000\) | \([2]\) | \(20901888\) | \(3.2516\) | |
212940.i1 | 212940bh4 | \([0, 0, 0, -100988823, 286573881022]\) | \(126449185587012304/33791748046875\) | \(30439552668985687500000000\) | \([2]\) | \(41803776\) | \(3.5981\) |
Rank
sage: E.rank()
The elliptic curves in class 212940bh have rank \(1\).
Complex multiplication
The elliptic curves in class 212940bh do not have complex multiplication.Modular form 212940.2.a.bh
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.