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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 30912e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
30912.v4 | 30912e1 | \([0, -1, 0, -38337, -4700223]\) | \(-23771111713777/22848457968\) | \(-5989586165563392\) | \([2]\) | \(184320\) | \(1.7236\) | \(\Gamma_0(N)\)-optimal |
30912.v3 | 30912e2 | \([0, -1, 0, -715457, -232618815]\) | \(154502321244119857/55101928644\) | \(14444639982452736\) | \([2, 2]\) | \(368640\) | \(2.0701\) | |
30912.v2 | 30912e3 | \([0, -1, 0, -818497, -161129663]\) | \(231331938231569617/90942310746882\) | \(23839981108430635008\) | \([2]\) | \(737280\) | \(2.4167\) | |
30912.v1 | 30912e4 | \([0, -1, 0, -11446337, -14901731775]\) | \(632678989847546725777/80515134\) | \(21106559287296\) | \([2]\) | \(737280\) | \(2.4167\) |
Rank
sage: E.rank()
The elliptic curves in class 30912e have rank \(0\).
Complex multiplication
The elliptic curves in class 30912e do not have complex multiplication.Modular form 30912.2.a.e
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.