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SageMath
E = EllipticCurve("ht1")
E.isogeny_class()
Elliptic curves in class 414960ht
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414960.ht4 | 414960ht1 | \([0, 1, 0, -5012600, -4039710252]\) | \(3400580030216483633401/248179914178560000\) | \(1016544928475381760000\) | \([2]\) | \(21233664\) | \(2.7772\) | \(\Gamma_0(N)\)-optimal |
414960.ht2 | 414960ht2 | \([0, 1, 0, -78740600, -268959159852]\) | \(13181351126943641326385401/78687520727961600\) | \(322304084901730713600\) | \([2, 2]\) | \(42467328\) | \(3.1237\) | |
414960.ht3 | 414960ht3 | \([0, 1, 0, -77281400, -279405280812]\) | \(-12462027714326806804452601/1020321931394362309440\) | \(-4179238630991308019466240\) | \([2]\) | \(84934656\) | \(3.4703\) | |
414960.ht1 | 414960ht4 | \([0, 1, 0, -1259847800, -17212178165292]\) | \(53990582156643221755293310201/1924038392640\) | \(7880861256253440\) | \([2]\) | \(84934656\) | \(3.4703\) |
Rank
sage: E.rank()
The elliptic curves in class 414960ht have rank \(0\).
Complex multiplication
The elliptic curves in class 414960ht do not have complex multiplication.Modular form 414960.2.a.ht
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.