Minimal Weierstrass equation
magma: E := EllipticCurve([0, 1, 1, -26790, -31917424]); // or
magma: E := EllipticCurve("1155n2");
magma: E := EllipticCurve("1155n2");
sage: E = EllipticCurve([0, 1, 1, -26790, -31917424]) # or
sage: E = EllipticCurve("1155n2")
sage: E = EllipticCurve("1155n2")
gp: E = ellinit([0, 1, 1, -26790, -31917424]) \\ or
gp: E = ellinit("1155n2")
gp: E = ellinit("1155n2")
\( y^2 + y = x^{3} + x^{2} - 26790 x - 31917424 \)
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
magma: Generators(E);
sage: E.gens()
| \(P\) | = | \( \left(\frac{6197}{4}, \frac{483149}{8}\right) \) |
| \(\hat{h}(P)\) | ≈ | 0.86625097826 |
Integral points
magma: IntegralPoints(E);
sage: E.integral_points()
None
Invariants
|
magma: Conductor(E);
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
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| Conductor: | \( 1155 \) | = | \(3 \cdot 5 \cdot 7 \cdot 11\) | ||
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magma: Discriminant(E);
sage: E.discriminant().factor()
gp: E.disc
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| Discriminant: | \(-438611057788643355 \) | = | \(-1 \cdot 3 \cdot 5 \cdot 7 \cdot 11^{15} \) | ||
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magma: jInvariant(E);
sage: E.j_invariant().factor()
gp: E.j
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| j-invariant: | \( -\frac{2126464142970105856}{438611057788643355} \) | = | \(-1 \cdot 2^{12} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{-1} \cdot 11^{-15} \cdot 179^{3} \cdot 449^{3}\) | ||
| Endomorphism ring: | \(\Z\) | (no Complex Multiplication) | |||
| Sato-Tate Group: | $\mathrm{SU}(2)$ | ||||
BSD invariants
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magma: Rank(E);
sage: E.rank()
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| Rank: | \(1\) | ||
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magma: Regulator(E);
sage: E.regulator()
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| Regulator: | \(0.86625097826\) | ||
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magma: RealPeriod(E);
sage: E.period_lattice().omega()
gp: E.omega[1]
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| Real period: | \(0.132670927494\) | ||
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magma: TamagawaNumbers(E);
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
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| Tamagawa product: | \( 15 \) = \( 1\cdot1\cdot1\cdot( 3 \cdot 5 ) \) | ||
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magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp: elltors(E)[1]
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| Torsion order: | \(1\) | ||
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magma: MordellWeilShaInformation(E);
sage: E.sha().an_numerical()
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| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
Modular form 1155.2.a.c
magma: ModularForm(E);
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
\( q - 2q^{2} + q^{3} + 2q^{4} + q^{5} - 2q^{6} + q^{7} + q^{9} - 2q^{10} + q^{11} + 2q^{12} - 6q^{13} - 2q^{14} + q^{15} - 4q^{16} - 7q^{17} - 2q^{18} - 5q^{19} + O(q^{20}) \)
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magma: ModularDegree(E);
sage: E.modular_degree()
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| Modular degree: | 30000 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 1 | ||
Special L-value
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
sage: r = E.rank();
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
gp: ar = ellanalyticrank(E);
gp: ar[2]/factorial(ar[1])
gp: ar[2]/factorial(ar[1])
\( L'(E,1) \) ≈ \( 1.72389481092 \)
Local data
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
sage: E.local_data()
gp: ellglobalred(E)[5]
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(3\) | \(1\) | \( I_{1} \) | Split multiplicative | -1 | 1 | 1 | 1 |
| \(5\) | \(1\) | \( I_{1} \) | Split multiplicative | -1 | 1 | 1 | 1 |
| \(7\) | \(1\) | \( I_{1} \) | Split multiplicative | -1 | 1 | 1 | 1 |
| \(11\) | \(15\) | \( I_{15} \) | Split multiplicative | -1 | 1 | 15 | 15 |
Galois representations
The 2-adic representation attached to this elliptic curve is surjective.
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
sage: rho = E.galois_representation();
sage: [rho.image_type(p) for p in rho.non_surjective()]
sage: [rho.image_type(p) for p in rho.non_surjective()]
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.
| prime | Image of Galois representation |
|---|---|
| \(5\) | B.1.2 |
$p$-adic data
$p$-adic regulators
sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]
\(p\)-adic regulators are not yet computed for curves that are not \(\Gamma_0\)-optimal.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ss | split | split | split | split | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary | ordinary |
| $\lambda$-invariant(s) | 8,1 | 4 | 6 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0,0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class 1155.c
consists of 2 curves linked by isogenies of
degree 5.
Growth of torsion in number fields
The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base-change curve |
|---|---|---|---|
| 3 | 3.1.4620.1 | \(\Z/2\Z\) | Not in database |
| 4 | \(\Q(\zeta_{5})\) | \(\Z/5\Z\) | Not in database |
| 5 | 5.1.379845703125.1 | \(\Z/5\Z\) | Not in database |
| 6 | 6.0.24652782000.1 | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.