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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 27489r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 27489.h3 | 27489r1 | \([1, 0, 0, -6556299, -6462078120]\) | \(264918160154242157473/536027170833\) | \(63063060621331617\) | \([2]\) | \(552960\) | \(2.4745\) | \(\Gamma_0(N)\)-optimal |
| 27489.h2 | 27489r2 | \([1, 0, 0, -6627104, -6315384321]\) | \(273594167224805799793/11903648120953281\) | \(1400452297782032556369\) | \([2, 2]\) | \(1105920\) | \(2.8211\) | |
| 27489.h4 | 27489r3 | \([1, 0, 0, 3373061, -23701671190]\) | \(36075142039228937567/2083708275110728497\) | \(-245146194858502096943553\) | \([2]\) | \(2211840\) | \(3.1677\) | |
| 27489.h1 | 27489r4 | \([1, 0, 0, -17760149, 20459588904]\) | \(5265932508006615127873/1510137598013239041\) | \(177666178268659559934609\) | \([2]\) | \(2211840\) | \(3.1677\) |
Rank
sage: E.rank()
The elliptic curves in class 27489r have rank \(1\).
Complex multiplication
The elliptic curves in class 27489r do not have complex multiplication.Modular form 27489.2.a.r
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
