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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 45486q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 45486.k6 | 45486q1 | \([1, -1, 0, -25343892, 49114758784]\) | \(52492168638015625/293197968\) | \(10055648643025990032\) | \([2]\) | \(3317760\) | \(2.8386\) | \(\Gamma_0(N)\)-optimal |
| 45486.k5 | 45486q2 | \([1, -1, 0, -25798752, 47260658452]\) | \(55369510069623625/3916046302812\) | \(134306475449033238364188\) | \([2]\) | \(6635520\) | \(3.1851\) | |
| 45486.k4 | 45486q3 | \([1, -1, 0, -36309267, 2597620405]\) | \(154357248921765625/89242711068672\) | \(3060707932524457664483328\) | \([2]\) | \(9953280\) | \(3.3879\) | |
| 45486.k3 | 45486q4 | \([1, -1, 0, -392919507, -2987721886091]\) | \(195607431345044517625/752875610010048\) | \(25820958643768307591957952\) | \([2]\) | \(19906560\) | \(3.7345\) | |
| 45486.k2 | 45486q5 | \([1, -1, 0, -1987317522, -34098857881772]\) | \(25309080274342544331625/191933498523648\) | \(6582637107432313011044352\) | \([2]\) | \(29859840\) | \(3.9372\) | |
| 45486.k1 | 45486q6 | \([1, -1, 0, -31797022482, -2182359093588140]\) | \(103665426767620308239307625/5961940992\) | \(204473394733778731008\) | \([2]\) | \(59719680\) | \(4.2838\) |
Rank
sage: E.rank()
The elliptic curves in class 45486q have rank \(0\).
Complex multiplication
The elliptic curves in class 45486q do not have complex multiplication.Modular form 45486.2.a.q
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
