Minimal Weierstrass equation
\(y^2+y=x^3+x^2-183x+919\) 
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
| \(P\) | = | \(\left(7, 5\right)\) |
| \(\hat{h}(P)\) | ≈ | $0.75377822559486338339955515665$ |
Integral points
\( \left(-11, 41\right) \), \( \left(-11, -42\right) \), \( \left(7, 5\right) \), \( \left(7, -6\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 3025 \) | = | \(5^{2} \cdot 11^{2}\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(-20796875 \) | = | \(-1 \cdot 5^{6} \cdot 11^{3} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -32768 \) | = | \(-1 \cdot 2^{15}\) |
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z[(1+\sqrt{-11})/2]\) | (potential complex multiplication) | |
| Sato-Tate group: | $N(\mathrm{U}(1))$ | ||
| Faltings height: | \(0.18164613188757573762920830647\dots\) | ||
| Stable Faltings height: | \(-1.2225466425290670856866572546\dots\) | ||
BSD invariants
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sage: E.rank()
magma: Rank(E);
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| Analytic rank: | \(1\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(0.75377822559486338339955515665\dots\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(2.1477081065172769177445555858\dots\) | ||
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sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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| Tamagawa product: | \( 2 \) = \( 1\cdot2 \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(1\) | ||
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sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
For more coefficients, see the Downloads section to the right.
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sage: E.modular_degree()
magma: ModularDegree(E);
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| Modular degree: | 432 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
\( L'(E,1) \) ≈ \( 3.2377912112525936753769633141331257185 \)
Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(5\) | \(1\) | \(I_0^{*}\) | Additive | 1 | 2 | 6 | 0 |
| \(11\) | \(2\) | \(III\) | Additive | 1 | 2 | 3 | 0 |
Galois representations
The mod \( p \) Galois representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois representation |
|---|---|
| \(11\) | B.10.3 |
For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -11 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -11 }{p}\right)=-1\).
$p$-adic data
$p$-adic regulators
Note: \(p\)-adic regulator data only exists for primes \(p\ge 5\) of good ordinary reduction.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ss | ordinary | add | ss | add | ss | ss | ss | ordinary | ss | ordinary | ordinary | ss | ss | ordinary |
| $\lambda$-invariant(s) | ? | 3 | - | 1,1 | - | 1,1 | 1,1 | 1,1 | 1 | 1,1 | 3 | 1 | 1,1 | 1,1 | 1 |
| $\mu$-invariant(s) | ? | 0 | - | 0,0 | - | 0,0 | 0,0 | 0,0 | 0 | 0,0 | 0 | 0 | 0,0 | 0,0 | 0 |
An entry ? indicates that the invariants have not yet been computed.
An entry - indicates that the invariants are not computed because the reduction is additive.
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
11.
Its isogeny class 3025.d
consists of 2 curves linked by isogenies of
degree 11.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 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 |
|---|---|---|---|
| $3$ | 3.1.44.1 | \(\Z/2\Z\) | Not in database |
| $4$ | 4.0.99825.1 | \(\Z/3\Z\) | Not in database |
| $4$ | 4.2.299475.1 | \(\Z/3\Z\) | Not in database |
| $6$ | 6.0.21296.1 | \(\Z/2\Z \times \Z/2\Z\) | Not in database |
| $8$ | 8.0.89685275625.4 | \(\Z/3\Z \times \Z/3\Z\) | Not in database |
| $8$ | 8.0.5536128125.1 | \(\Z/5\Z\) | Not in database |
| $10$ | 10.10.669871503125.1 | \(\Z/11\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/4\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/9\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/2\Z \times \Z/6\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/6\Z\) | Not in database |
| $16$ | 16.4.766217865410400390625.1 | \(\Z/5\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/15\Z\) | Not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.