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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 2450.t
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2450.t1 | 2450y6 | \([1, 0, 0, -3344888, 2354339392]\) | \(2251439055699625/25088\) | \(46118408000000\) | \([2]\) | \(41472\) | \(2.1908\) | |
2450.t2 | 2450y5 | \([1, 0, 0, -208888, 36835392]\) | \(-548347731625/1835008\) | \(-3373232128000000\) | \([2]\) | \(20736\) | \(1.8442\) | |
2450.t3 | 2450y4 | \([1, 0, 0, -43513, 2860017]\) | \(4956477625/941192\) | \(1730160900125000\) | \([2]\) | \(13824\) | \(1.6415\) | |
2450.t4 | 2450y2 | \([1, 0, 0, -12888, -563858]\) | \(128787625/98\) | \(180150031250\) | \([2]\) | \(4608\) | \(1.0922\) | |
2450.t5 | 2450y1 | \([1, 0, 0, -638, -12608]\) | \(-15625/28\) | \(-51471437500\) | \([2]\) | \(2304\) | \(0.74559\) | \(\Gamma_0(N)\)-optimal |
2450.t6 | 2450y3 | \([1, 0, 0, 5487, 263017]\) | \(9938375/21952\) | \(-40353607000000\) | \([2]\) | \(6912\) | \(1.2949\) |
Rank
sage: E.rank()
The elliptic curves in class 2450.t have rank \(1\).
Complex multiplication
The elliptic curves in class 2450.t do not have complex multiplication.Modular form 2450.2.a.t
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.