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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 85008.cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 85008.cf1 | 85008ck4 | \([0, 1, 0, -153373792, -731146953100]\) | \(97413070452067229637409633/140666577176907936\) | \(576170300116614905856\) | \([2]\) | \(8847360\) | \(3.2543\) | |
| 85008.cf2 | 85008ck3 | \([0, 1, 0, -24569952, 31617312372]\) | \(400476194988122984445793/126270124548858769248\) | \(517202430152125518839808\) | \([4]\) | \(8847360\) | \(3.2543\) | |
| 85008.cf3 | 85008ck2 | \([0, 1, 0, -9673312, -11207548300]\) | \(24439335640029940889953/902916953746891776\) | \(3698347842547268714496\) | \([2, 2]\) | \(4423680\) | \(2.9077\) | |
| 85008.cf4 | 85008ck1 | \([0, 1, 0, 239008, -625155468]\) | \(368637286278891167/41443067603976192\) | \(-169750804905886482432\) | \([2]\) | \(2211840\) | \(2.5611\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 85008.cf have rank \(1\).
Complex multiplication
The elliptic curves in class 85008.cf do not have complex multiplication.Modular form 85008.2.a.cf
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
