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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 123210bi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 123210.c8 | 123210bi1 | \([1, -1, 0, 18225, -2912355]\) | \(357911/2160\) | \(-4040095432667760\) | \([2]\) | \(829440\) | \(1.6765\) | \(\Gamma_0(N)\)-optimal |
| 123210.c6 | 123210bi2 | \([1, -1, 0, -228195, -37953279]\) | \(702595369/72900\) | \(136353220852536900\) | \([2, 2]\) | \(1658880\) | \(2.0231\) | |
| 123210.c7 | 123210bi3 | \([1, -1, 0, -166590, 85687956]\) | \(-273359449/1536000\) | \(-2872956752119296000\) | \([2]\) | \(2488320\) | \(2.2258\) | |
| 123210.c5 | 123210bi4 | \([1, -1, 0, -844245, 257381091]\) | \(35578826569/5314410\) | \(9940149800149940010\) | \([2]\) | \(3317760\) | \(2.3696\) | |
| 123210.c4 | 123210bi5 | \([1, -1, 0, -3554865, -2578863825]\) | \(2656166199049/33750\) | \(63126491135433750\) | \([2]\) | \(3317760\) | \(2.3696\) | |
| 123210.c3 | 123210bi6 | \([1, -1, 0, -4109310, 3201225300]\) | \(4102915888729/9000000\) | \(16833730969449000000\) | \([2, 2]\) | \(4976640\) | \(2.5724\) | |
| 123210.c1 | 123210bi7 | \([1, -1, 0, -65714310, 205056168300]\) | \(16778985534208729/81000\) | \(151503578725041000\) | \([2]\) | \(9953280\) | \(2.9189\) | |
| 123210.c2 | 123210bi8 | \([1, -1, 0, -5587830, 693359676]\) | \(10316097499609/5859375000\) | \(10959460266568359375000\) | \([2]\) | \(9953280\) | \(2.9189\) |
Rank
sage: E.rank()
The elliptic curves in class 123210bi have rank \(0\).
Complex multiplication
The elliptic curves in class 123210bi do not have complex multiplication.Modular form 123210.2.a.bi
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.
