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SageMath
E = EllipticCurve("ej1")
E.isogeny_class()
Elliptic curves in class 257400.ej
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
257400.ej1 | 257400ej4 | \([0, 0, 0, -5773692675, -168860709049250]\) | \(912446049969377120252018/17177299425\) | \(400712040986400000000\) | \([2]\) | \(132120576\) | \(3.9414\) | |
257400.ej2 | 257400ej3 | \([0, 0, 0, -393042675, -2139843199250]\) | \(287849398425814280018/81784533026485575\) | \(1907869586441855493600000000\) | \([2]\) | \(132120576\) | \(3.9414\) | |
257400.ej3 | 257400ej2 | \([0, 0, 0, -360867675, -2638266124250]\) | \(445574312599094932036/61129333175625\) | \(713012542160490000000000\) | \([2, 2]\) | \(66060288\) | \(3.5948\) | |
257400.ej4 | 257400ej1 | \([0, 0, 0, -20555175, -48828311750]\) | \(-329381898333928144/162600887109375\) | \(-474144186810937500000000\) | \([2]\) | \(33030144\) | \(3.2482\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 257400.ej have rank \(1\).
Complex multiplication
The elliptic curves in class 257400.ej do not have complex multiplication.Modular form 257400.2.a.ej
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.