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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 12342k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
12342.p4 | 12342k1 | \([1, 0, 1, -707490, 179061364]\) | \(22106889268753393/4969545596928\) | \(8803853167239364608\) | \([2]\) | \(322560\) | \(2.3481\) | \(\Gamma_0(N)\)-optimal |
12342.p2 | 12342k2 | \([1, 0, 1, -10619810, 13318832756]\) | \(74768347616680342513/5615307472896\) | \(9947859721991110656\) | \([2, 2]\) | \(645120\) | \(2.6946\) | |
12342.p1 | 12342k3 | \([1, 0, 1, -169913890, 852480046196]\) | \(306234591284035366263793/1727485056\) | \(3060345153292416\) | \([2]\) | \(1290240\) | \(3.0412\) | |
12342.p3 | 12342k4 | \([1, 0, 1, -9922850, 15142637684]\) | \(-60992553706117024753/20624795251201152\) | \(-36538082900013164038272\) | \([2]\) | \(1290240\) | \(3.0412\) |
Rank
sage: E.rank()
The elliptic curves in class 12342k have rank \(1\).
Complex multiplication
The elliptic curves in class 12342k do not have complex multiplication.Modular form 12342.2.a.k
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.