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SageMath
E = EllipticCurve("ec1")
E.isogeny_class()
Elliptic curves in class 248430.ec
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 248430.ec1 | 248430ec5 | \([1, 0, 1, -139120973, 631580989106]\) | \(524388516989299201/3150\) | \(1788788143929150\) | \([2]\) | \(28311552\) | \(2.9922\) | |
| 248430.ec2 | 248430ec3 | \([1, 0, 1, -8695223, 9867524006]\) | \(128031684631201/9922500\) | \(5634682653376822500\) | \([2, 2]\) | \(14155776\) | \(2.6456\) | |
| 248430.ec3 | 248430ec6 | \([1, 0, 1, -8115553, 11239950698]\) | \(-104094944089921/35880468750\) | \(-20375414951942974218750\) | \([2]\) | \(28311552\) | \(2.9922\) | |
| 248430.ec4 | 248430ec4 | \([1, 0, 1, -3064143, -1951516682]\) | \(5602762882081/345888060\) | \(196419193922112530460\) | \([2]\) | \(14155776\) | \(2.6456\) | |
| 248430.ec5 | 248430ec2 | \([1, 0, 1, -579843, 132314158]\) | \(37966934881/8643600\) | \(4908434666941587600\) | \([2, 2]\) | \(7077888\) | \(2.2990\) | |
| 248430.ec6 | 248430ec1 | \([1, 0, 1, 82637, 12802766]\) | \(109902239/188160\) | \(-106850278464034560\) | \([2]\) | \(3538944\) | \(1.9525\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 248430.ec have rank \(2\).
Complex multiplication
The elliptic curves in class 248430.ec do not have complex multiplication.Modular form 248430.2.a.ec
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
