This is the modular curve $X_{\text{ns}}^{+}(11)$.
Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3-x^2-7x+10\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3-x^2z-7xz^2+10z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-9504x+365904\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4, 5)$ | $0.089785156160690453326303090165$ | $\infty$ |
Integral points
\( \left(-2, 3\right) \), \( \left(-2, -4\right) \), \( \left(2, 0\right) \), \( \left(2, -1\right) \), \( \left(4, 5\right) \), \( \left(4, -6\right) \)
Invariants
| Conductor: | $N$ | = | \( 121 \) | = | $11^{2}$ |
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| Discriminant: | $\Delta$ | = | $-1331$ | = | $-1 \cdot 11^{3} $ |
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| j-invariant: | $j$ | = | \( -32768 \) | = | $-1 \cdot 2^{15}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-11})/2]\) (potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.62307282432947444967117136014$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.2225466425290670856866572546$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0251241218312794$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.6787020419109124$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.089785156160690453326303090165$ |
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| Real period: | $\Omega$ | ≈ | $4.8024213219999902955416548663$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $0.86237229669039723995955162179 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 0.862372297 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 4.802421 \cdot 0.089785 \cdot 2}{1^2} \\ & \approx 0.862372297\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There is only one prime $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $11$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
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 |
|---|---|---|
| $11$ | 11B.1.3 | 11.120.1.5 |
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $11$ | additive | $42$ | \( 1 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
11.
Its isogeny class 121.b
consists of 2 curves linked by isogenies of
degree 11.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
| $4$ | 4.0.3993.1 | \(\Z/3\Z\) | not in database |
| $4$ | 4.2.11979.1 | \(\Z/3\Z\) | not in database |
| $5$ | \(\Q(\zeta_{11})^+\) | \(\Z/11\Z\) | 5.5.14641.1-121.1-b1 |
| $6$ | 6.0.21296.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.0.143496441.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $8$ | 8.0.221445125.1 | \(\Z/5\Z\) | not in database |
| $12$ | 12.2.2472487358038016.1 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/9\Z\) | not in database |
| $12$ | 12.0.16298134440192.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.2.440049629885184.1 | \(\Z/6\Z\) | not in database |
| $15$ | 15.5.35351257235385344.1 | \(\Z/22\Z\) | not in database |
| $16$ | 16.4.766217865410400390625.1 | \(\Z/5\Z\) | not in database |
| $16$ | deg 16 | \(\Z/15\Z\) | not in database |
| $20$ | 20.0.14861658978964734748713.1 | \(\Z/33\Z\) | not in database |
| $20$ | 20.10.3611383131888430543937259.1 | \(\Z/33\Z\) | 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.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ss | ord | ord | ss | add | ss | ss | ss | ord | ss | ord | ord | ss | ss | ord |
| $\lambda$-invariant(s) | ? | 1 | 1 | 1,1 | - | 1,1 | 1,1 | 1,1 | 1 | 1,1 | 1 | 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 | 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.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.
Additional information
For the identification of this curve $E$ with the modular curve $X_{\text{nonsplit}}(11)$, see [E. Halberstadt's paper] "Sur la courbe modulaire $X_{ndép}(11)$" ["On the modular curve $X_{\text{nonsplit}}(11)$"], Experimental Math. 7 (1998) #2, 163-174. This modular curve classifies (up to twist) elliptic curves $\mathcal E$ such that the Galois action on $\mathcal{E}[11]$ factors through the normalizer of a non-split Cartan subgroup of $\text{GL}_2({\bf F}_{11})$. Since $E(\Q)$ is infinite, there are infinitely many $j$-invariants of such $\mathcal E$. Halberstadt computes the first few examples; most are CM curves, and of the non-CM curves the first conductor (and the oniy one within the LMFDB range $N \leq 4\cdot 10^5$) is $232544 = 2^5 13^2 43$, arising for the curves [232544.f1], [232544.g1], [232544.h1], and [232544.i1] which are related by quadratic twists by $\Q(\sqrt{-1})$ and $\Q(\sqrt{\pm13})$.