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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 224400cr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 224400.hn3 | 224400cr1 | \([0, 1, 0, -82179208, 283819001588]\) | \(959024269496848362625/11151660319506432\) | \(713706260448411648000000\) | \([2]\) | \(39813120\) | \(3.3895\) | \(\Gamma_0(N)\)-optimal |
| 224400.hn4 | 224400cr2 | \([0, 1, 0, -16643208, 724089849588]\) | \(-7966267523043306625/3534510366354604032\) | \(-226208663446694658048000000\) | \([2]\) | \(79626240\) | \(3.7361\) | |
| 224400.hn1 | 224400cr3 | \([0, 1, 0, -6637827208, 208152873977588]\) | \(505384091400037554067434625/815656731648\) | \(52202030825472000000\) | \([2]\) | \(119439360\) | \(3.9388\) | |
| 224400.hn2 | 224400cr4 | \([0, 1, 0, -6637763208, 208157088633588]\) | \(-505369473241574671219626625/20303219722982711328\) | \(-1299406062270893524992000000\) | \([2]\) | \(238878720\) | \(4.2854\) |
Rank
sage: E.rank()
The elliptic curves in class 224400cr have rank \(0\).
Complex multiplication
The elliptic curves in class 224400cr do not have complex multiplication.Modular form 224400.2.a.cr
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.
