Show commands:
SageMath
E = EllipticCurve("hz1")
E.isogeny_class()
Elliptic curves in class 162624hz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
162624.cf4 | 162624hz1 | \([0, -1, 0, 3227, 22981]\) | \(2048000/1323\) | \(-2400025807872\) | \([2]\) | \(207360\) | \(1.0645\) | \(\Gamma_0(N)\)-optimal |
162624.cf3 | 162624hz2 | \([0, -1, 0, -13713, 202545]\) | \(9826000/5103\) | \(148115878428672\) | \([2]\) | \(414720\) | \(1.4110\) | |
162624.cf2 | 162624hz3 | \([0, -1, 0, -54853, 5110789]\) | \(-10061824000/352947\) | \(-640273551633408\) | \([2]\) | \(622080\) | \(1.6138\) | |
162624.cf1 | 162624hz4 | \([0, -1, 0, -884913, 320699601]\) | \(2640279346000/3087\) | \(89600963493888\) | \([2]\) | \(1244160\) | \(1.9603\) |
Rank
sage: E.rank()
The elliptic curves in class 162624hz have rank \(0\).
Complex multiplication
The elliptic curves in class 162624hz do not have complex multiplication.Modular form 162624.2.a.hz
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.