The first elliptic curve in nature. This is a model for the modular curve \(X_1(11)\).
Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3-x^2\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3-x^2z\)
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(dehomogenize, simplify) |
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\(y^2=x^3-432x+8208\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{5}\Z\)
Torsion generators
\( \left(0, 0\right) \)
Integral points
\( \left(0, 0\right) \), \( \left(0, -1\right) \), \( \left(1, 0\right) \), \( \left(1, -1\right) \)
Invariants
| Conductor: | \( 11 \) | = | $11$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-11 $ | = | $-1 \cdot 11 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( -\frac{4096}{11} \) | = | $-1 \cdot 2^{12} \cdot 11^{-1}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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| Endomorphism ring: | $\Z$ | |||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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| Sato-Tate group: | $\mathrm{SU}(2)$ | |||
| Faltings height: | $-1.1127287973354532519893939281\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $-1.1127287973354532519893939281\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
| Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $6.3460465213977671084439730838\dots$ | comment: Real Period
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
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| Tamagawa product: | $ 1 $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
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| Torsion order: | $5$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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| Analytic order of Ш: | $1$ (exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Special value: | $ L(E,1) $ ≈ $ 0.25384186085591068433775892335 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 0.253841861 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 6.346047 \cdot 1.000000 \cdot 1}{5^2} \approx 0.253841861$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 5 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There is only one prime of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $11$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $5$ | 5B.1.1 | 25.120.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 550 = 2 \cdot 5^{2} \cdot 11 \), index $1200$, genus $37$, and generators
$\left(\begin{array}{rr} 316 & 545 \\ 377 & 287 \end{array}\right),\left(\begin{array}{rr} 38 & 41 \\ 191 & 539 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 50 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 501 & 50 \\ 500 & 51 \end{array}\right),\left(\begin{array}{rr} 516 & 5 \\ 35 & 186 \end{array}\right)$.
The torsion field $K:=\Q(E[550])$ is a degree-$19800000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/550\Z)$.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 11 |
|---|---|---|---|---|
| Reduction type | ss | ord | ord | split |
| $\lambda$-invariant(s) | 0,1 | 0 | 0 | 1 |
| $\mu$-invariant(s) | 0,0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5 and 25.
Its isogeny class 11.a
consists of 3 curves linked by isogenies of
degrees dividing 25.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{5}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.44.1 | \(\Z/10\Z\) | Not in database |
| $5$ | \(\Q(\zeta_{11})^+\) | \(\Z/25\Z\) | 5.5.14641.1-11.1-a3 |
| $6$ | 6.0.21296.1 | \(\Z/2\Z \oplus \Z/10\Z\) | Not in database |
| $8$ | 8.2.32019867.1 | \(\Z/15\Z\) | Not in database |
| $12$ | 12.2.20433779818496.3 | \(\Z/20\Z\) | Not in database |
| $15$ | 15.5.35351257235385344.1 | \(\Z/50\Z\) | Not in database |
| $20$ | 20.0.1402274470934209014892578125.2 | \(\Z/5\Z \oplus \Z/5\Z\) | Not in database |
| $20$ | 20.0.95777233176300048828125.1 | \(\Z/25\Z\) | Not in database |
We only show fields where the torsion growth is primitive.