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SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 4410bb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4410.z8 | 4410bb1 | \([1, -1, 1, 652, 19527]\) | \(357911/2160\) | \(-185254821360\) | \([2]\) | \(4608\) | \(0.84397\) | \(\Gamma_0(N)\)-optimal |
| 4410.z6 | 4410bb2 | \([1, -1, 1, -8168, 259431]\) | \(702595369/72900\) | \(6252350220900\) | \([2, 2]\) | \(9216\) | \(1.1905\) | |
| 4410.z7 | 4410bb3 | \([1, -1, 1, -5963, -578469]\) | \(-273359449/1536000\) | \(-131736761856000\) | \([2]\) | \(13824\) | \(1.3933\) | |
| 4410.z5 | 4410bb4 | \([1, -1, 1, -30218, -1733889]\) | \(35578826569/5314410\) | \(455796331103610\) | \([2]\) | \(18432\) | \(1.5371\) | |
| 4410.z4 | 4410bb5 | \([1, -1, 1, -127238, 17500767]\) | \(2656166199049/33750\) | \(2894606583750\) | \([2]\) | \(18432\) | \(1.5371\) | |
| 4410.z3 | 4410bb6 | \([1, -1, 1, -147083, -21633573]\) | \(4102915888729/9000000\) | \(771895089000000\) | \([2, 2]\) | \(27648\) | \(1.7399\) | |
| 4410.z1 | 4410bb7 | \([1, -1, 1, -2352083, -1387851573]\) | \(16778985534208729/81000\) | \(6947055801000\) | \([2]\) | \(55296\) | \(2.0864\) | |
| 4410.z2 | 4410bb8 | \([1, -1, 1, -200003, -4635669]\) | \(10316097499609/5859375000\) | \(502535865234375000\) | \([2]\) | \(55296\) | \(2.0864\) |
Rank
sage: E.rank()
The elliptic curves in class 4410bb have rank \(0\).
Complex multiplication
The elliptic curves in class 4410bb do not have complex multiplication.Modular form 4410.2.a.bb
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
