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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 340032.w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 340032.w1 | 340032w4 | \([0, -1, 0, -613495169, -5848562129631]\) | \(97413070452067229637409633/140666577176907936\) | \(36874899207463353974784\) | \([2]\) | \(70778880\) | \(3.6008\) | |
| 340032.w2 | 340032w3 | \([0, -1, 0, -98279809, 253036778785]\) | \(400476194988122984445793/126270124548858769248\) | \(33100955529736033205747712\) | \([2]\) | \(70778880\) | \(3.6008\) | |
| 340032.w3 | 340032w2 | \([0, -1, 0, -38693249, -89621693151]\) | \(24439335640029940889953/902916953746891776\) | \(236694261923025197727744\) | \([2, 2]\) | \(35389440\) | \(3.2543\) | |
| 340032.w4 | 340032w1 | \([0, -1, 0, 956031, -5002199775]\) | \(368637286278891167/41443067603976192\) | \(-10864051513976734875648\) | \([2]\) | \(17694720\) | \(2.9077\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 340032.w have rank \(1\).
Complex multiplication
The elliptic curves in class 340032.w do not have complex multiplication.Modular form 340032.2.a.w
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.
