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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 1254.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1254.f1 | 1254g5 | \([1, 1, 1, -148049, -21987223]\) | \(358872624127382648977/5938169721462\) | \(5938169721462\) | \([2]\) | \(6144\) | \(1.5823\) | |
1254.f2 | 1254g3 | \([1, 1, 1, -9539, -324259]\) | \(95992014075197617/11235515171364\) | \(11235515171364\) | \([2, 2]\) | \(3072\) | \(1.2357\) | |
1254.f3 | 1254g2 | \([1, 1, 1, -2319, 36741]\) | \(1379233073341297/183927761424\) | \(183927761424\) | \([2, 4]\) | \(1536\) | \(0.88916\) | |
1254.f4 | 1254g1 | \([1, 1, 1, -2239, 39845]\) | \(1241361053832817/27447552\) | \(27447552\) | \([4]\) | \(768\) | \(0.54259\) | \(\Gamma_0(N)\)-optimal |
1254.f5 | 1254g4 | \([1, 1, 1, 3621, 200685]\) | \(5250513632788943/20176472892708\) | \(-20176472892708\) | \([4]\) | \(3072\) | \(1.2357\) | |
1254.f6 | 1254g6 | \([1, 1, 1, 13451, -1620895]\) | \(269144439804255023/1298611008739638\) | \(-1298611008739638\) | \([2]\) | \(6144\) | \(1.5823\) |
Rank
sage: E.rank()
The elliptic curves in class 1254.f have rank \(1\).
Complex multiplication
The elliptic curves in class 1254.f do not have complex multiplication.Modular form 1254.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.