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SageMath
E = EllipticCurve("do1")
E.isogeny_class()
Elliptic curves in class 318402do
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
318402.do2 | 318402do1 | \([1, -1, 1, -2471015105, 47278846173521]\) | \(11165451838341046875/572244736\) | \(85517582701982832269568\) | \([2]\) | \(159252480\) | \(3.8732\) | \(\Gamma_0(N)\)-optimal |
318402.do3 | 318402do2 | \([1, -1, 1, -2466769745, 47449392473585]\) | \(-11108001800138902875/79947274872976\) | \(-11947506478675374152497856688\) | \([2]\) | \(318504960\) | \(4.2198\) | |
318402.do1 | 318402do3 | \([1, -1, 1, -2692039160, 38319083977339]\) | \(19804628171203875/5638671302656\) | \(614296324889388751092461862912\) | \([2]\) | \(477757440\) | \(4.4225\) | |
318402.do4 | 318402do4 | \([1, -1, 1, 7089270280, 252772337187451]\) | \(361682234074684125/462672528510976\) | \(-50405143097753797544779443351552\) | \([2]\) | \(955514880\) | \(4.7691\) |
Rank
sage: E.rank()
The elliptic curves in class 318402do have rank \(0\).
Complex multiplication
The elliptic curves in class 318402do do not have complex multiplication.Modular form 318402.2.a.do
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.