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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 37830bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 37830.z6 | 37830bd1 | \([1, 0, 0, -861900, 307915920]\) | \(70809965983288820433601/2711034593280\) | \(2711034593280\) | \([4]\) | \(319488\) | \(1.8769\) | \(\Gamma_0(N)\)-optimal |
| 37830.z5 | 37830bd2 | \([1, 0, 0, -863180, 306955152]\) | \(71125912466869080150721/438076768276742400\) | \(438076768276742400\) | \([2, 4]\) | \(638976\) | \(2.2234\) | |
| 37830.z7 | 37830bd3 | \([1, 0, 0, -358780, 662960672]\) | \(-5107501047547200669121/186932676651373720080\) | \(-186932676651373720080\) | \([4]\) | \(1277952\) | \(2.5700\) | |
| 37830.z4 | 37830bd4 | \([1, 0, 0, -1388060, -110534400]\) | \(295766137257618460155841/165893887376396010000\) | \(165893887376396010000\) | \([2, 4]\) | \(1277952\) | \(2.5700\) | |
| 37830.z8 | 37830bd5 | \([1, 0, 0, 5456440, -875749500]\) | \(17966019185011323998732159/10728662847342621912900\) | \(-10728662847342621912900\) | \([4]\) | \(2555904\) | \(2.9166\) | |
| 37830.z2 | 37830bd6 | \([1, 0, 0, -16630640, -26062551108]\) | \(508686956122817563500015361/971393810414076562500\) | \(971393810414076562500\) | \([2, 2]\) | \(2555904\) | \(2.9166\) | |
| 37830.z3 | 37830bd7 | \([1, 0, 0, -11174390, -43447254858]\) | \(-154310397026165916426315361/726135874771145001491250\) | \(-726135874771145001491250\) | \([2]\) | \(5111808\) | \(3.2631\) | |
| 37830.z1 | 37830bd8 | \([1, 0, 0, -265968170, -1669545946350]\) | \(2080715524939479717643838309281/7609177551269531250\) | \(7609177551269531250\) | \([2]\) | \(5111808\) | \(3.2631\) |
Rank
sage: E.rank()
The elliptic curves in class 37830bd have rank \(0\).
Complex multiplication
The elliptic curves in class 37830bd do not have complex multiplication.Modular form 37830.2.a.bd
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
