Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-23x+168\)
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(homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-23xz^2+168z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-363x+10406\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
$P$ | $\hat{h}(P)$ | Order |
---|---|---|
$(14, 42)$ | $0.13276237501049809530571116873$ | $\infty$ |
Integral points
\( \left(-4, 15\right) \), \( \left(-4, -12\right) \), \( \left(3, 9\right) \), \( \left(3, -13\right) \), \( \left(14, 42\right) \), \( \left(14, -57\right) \)
Invariants
Conductor: | $N$ | = | \( 1089 \) | = | $3^{2} \cdot 11^{2}$ |
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Discriminant: | $\Delta$ | = | $-10673289$ | = | $-1 \cdot 3^{6} \cdot 11^{4} $ |
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j-invariant: | $j$ | = | \( -121 \) | = | $-1 \cdot 11^{2}$ |
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Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $0.031345306218375756312683160611$ |
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Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.3172592623818026040722539838$ |
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$abc$ quality: | $Q$ | ≈ | $0.9461121308337243$ | |||
Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.3899047317728104$ |
BSD invariants
Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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Mordell-Weil rank: | $r$ | = | $ 1$ |
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Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.13276237501049809530571116873$ |
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Real period: | $\Omega$ | ≈ | $1.9527584900431847587163473241$ |
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Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 2\cdot3 $ |
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Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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Special value: | $ L'(E,1)$ | ≈ | $1.5555171297602838352109652600 $ |
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Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.555517130 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.952758 \cdot 0.132762 \cdot 6}{1^2} \\ & \approx 1.555517130\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 144 |
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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 are 2 primes $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
$11$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 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 |
---|---|---|
$2$ | 2G | 4.2.0.1 |
$11$ | 11B.10.4 | 11.60.1.4 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 264 = 2^{3} \cdot 3 \cdot 11 \), index $480$, genus $16$, and generators
$\left(\begin{array}{rr} 73 & 0 \\ 0 & 49 \end{array}\right),\left(\begin{array}{rr} 177 & 88 \\ 176 & 177 \end{array}\right),\left(\begin{array}{rr} 133 & 132 \\ 66 & 1 \end{array}\right),\left(\begin{array}{rr} 175 & 0 \\ 0 & 263 \end{array}\right),\left(\begin{array}{rr} 133 & 33 \\ 0 & 67 \end{array}\right),\left(\begin{array}{rr} 133 & 132 \\ 132 & 133 \end{array}\right),\left(\begin{array}{rr} 1 & 132 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 88 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 132 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 84 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 100 & 231 \\ 165 & 199 \end{array}\right),\left(\begin{array}{rr} 133 & 198 \\ 198 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[264])$ is a degree-$2027520$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/264\Z)$.
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 |
---|---|---|---|
$3$ | additive | $6$ | \( 121 = 11^{2} \) |
$11$ | additive | $52$ | \( 9 = 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
11.
Its isogeny class 1089.c
consists of 2 curves linked by isogenies of
degree 11.
Twists
The minimal quadratic twist of this elliptic curve is 121.c2, its twist by $-3$.
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 |
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$3$ | 3.1.484.1 | \(\Z/2\Z\) | not in database |
$6$ | 6.0.937024.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$8$ | 8.2.32019867.2 | \(\Z/3\Z\) | not in database |
$10$ | \(\Q(\zeta_{33})^+\) | \(\Z/11\Z\) | not in database |
$12$ | deg 12 | \(\Z/4\Z\) | 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 |
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Reduction type | ord | add | ord | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ss | ord |
$\lambda$-invariant(s) | ? | - | 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 |
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.