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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 446160.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
446160.a1 | 446160a8 | \([0, -1, 0, -2831088056, 32598851400816]\) | \(126929854754212758768001/50235797102795981820\) | \(993192335679281439590860308480\) | \([2]\) | \(668860416\) | \(4.4543\) | |
446160.a2 | 446160a6 | \([0, -1, 0, -2471185656, 47269049068656]\) | \(84415028961834287121601/30783551683856400\) | \(608609584413094814628249600\) | \([2, 2]\) | \(334430208\) | \(4.1077\) | |
446160.a3 | 446160a3 | \([0, -1, 0, -2470969336, 47277740633200]\) | \(84392862605474684114881/11228954880\) | \(222003283867147960320\) | \([2]\) | \(167215104\) | \(3.7611\) | \(\Gamma_0(N)\)-optimal* |
446160.a4 | 446160a7 | \([0, -1, 0, -2114744376, 61382983144560]\) | \(-52902632853833942200321/51713453577420277500\) | \(-1022406505056560710531061760000\) | \([2]\) | \(668860416\) | \(4.4543\) | |
446160.a5 | 446160a5 | \([0, -1, 0, -1276017656, -17542015046544]\) | \(11621808143080380273601/1335706803288000\) | \(26407737833356279775232000\) | \([2]\) | \(222953472\) | \(3.9050\) | |
446160.a6 | 446160a2 | \([0, -1, 0, -86257656, -226723910544]\) | \(3590017885052913601/954068544000000\) | \(18862516776124809216000000\) | \([2, 2]\) | \(111476736\) | \(3.5584\) | |
446160.a7 | 446160a1 | \([0, -1, 0, -30879736, 63190576240]\) | \(164711681450297281/8097103872000\) | \(160084680062175019008000\) | \([2]\) | \(55738368\) | \(3.2118\) | \(\Gamma_0(N)\)-optimal* |
446160.a8 | 446160a4 | \([0, -1, 0, 217455624, -1466845975440]\) | \(57519563401957999679/80296734375000000\) | \(-1587515392622016000000000000\) | \([2]\) | \(222953472\) | \(3.9050\) |
Rank
sage: E.rank()
The elliptic curves in class 446160.a have rank \(0\).
Complex multiplication
The elliptic curves in class 446160.a do not have complex multiplication.Modular form 446160.2.a.a
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 3 & 6 & 12 & 12 \\ 2 & 1 & 2 & 2 & 6 & 3 & 6 & 6 \\ 4 & 2 & 1 & 4 & 12 & 6 & 3 & 12 \\ 4 & 2 & 4 & 1 & 12 & 6 & 12 & 3 \\ 3 & 6 & 12 & 12 & 1 & 2 & 4 & 4 \\ 6 & 3 & 6 & 6 & 2 & 1 & 2 & 2 \\ 12 & 6 & 3 & 12 & 4 & 2 & 1 & 4 \\ 12 & 6 & 12 & 3 & 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.