Show commands:
SageMath
E = EllipticCurve("di1")
E.isogeny_class()
Elliptic curves in class 134640di
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
134640.fq4 | 134640di1 | \([0, 0, 0, -67347, 534085586]\) | \(-305460292990923/1114070936704000\) | \(-123207333031968768000\) | \([2]\) | \(3870720\) | \(2.5340\) | \(\Gamma_0(N)\)-optimal |
134640.fq3 | 134640di2 | \([0, 0, 0, -10289427, 12540940754]\) | \(1089365384367428097483/16063552169500000\) | \(1776500361529344000000\) | \([2]\) | \(7741440\) | \(2.8805\) | |
134640.fq2 | 134640di3 | \([0, 0, 0, -51903747, 143935454466]\) | \(-191808834096148160787/11043434659840\) | \(-890339018381847429120\) | \([2]\) | \(11612160\) | \(3.0833\) | |
134640.fq1 | 134640di4 | \([0, 0, 0, -830471427, 9211601796354]\) | \(785681552361835673854227/2604236800\) | \(209957654259302400\) | \([2]\) | \(23224320\) | \(3.4299\) |
Rank
sage: E.rank()
The elliptic curves in class 134640di have rank \(0\).
Complex multiplication
The elliptic curves in class 134640di do not have complex multiplication.Modular form 134640.2.a.di
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.