Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+10694401x+17998163481\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+10694401xz^2+17998163481z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+13859943669x+839680735538502\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 122694 \) | = | $2 \cdot 3 \cdot 11^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-218205319995451521368064$ | = | $-1 \cdot 2^{24} \cdot 3^{2} \cdot 11^{6} \cdot 13^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{93603087383}{150994944} \) | = | $2^{-24} \cdot 3^{-2} \cdot 13 \cdot 1931^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1634554360921960482676248012$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.25454156138531961886766138451$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0580983204947527$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.184585742645341$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.068018840276231499463168043896$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ ( 2^{3} \cdot 3 )\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $6.5298086665182239484641322140 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 6.529808667 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.068019 \cdot 1.000000 \cdot 96}{1^2} \\ & \approx 6.529808667\end{aligned}$$
Modular invariants
Modular form 122694.2.a.dh
For more coefficients, see the Downloads section to the right.
| Modular degree: | 16174080 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $24$ | $I_{24}$ | split multiplicative | -1 | 1 | 24 | 24 |
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $11$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $1$ | $IV^{*}$ | additive | 1 | 2 | 8 | 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 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 264 = 2^{3} \cdot 3 \cdot 11 \), index $32$, genus $0$, and generators
$\left(\begin{array}{rr} 87 & 187 \\ 209 & 164 \end{array}\right),\left(\begin{array}{rr} 119 & 0 \\ 0 & 263 \end{array}\right),\left(\begin{array}{rr} 253 & 12 \\ 252 & 13 \end{array}\right),\left(\begin{array}{rr} 5 & 2 \\ 226 & 249 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 221 & 33 \\ 187 & 100 \end{array}\right),\left(\begin{array}{rr} 4 & 9 \\ 3 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 67 & 132 \\ 33 & 1 \end{array}\right),\left(\begin{array}{rr} 133 & 33 \\ 231 & 100 \end{array}\right)$.
The torsion field $K:=\Q(E[264])$ is a degree-$30412800$ 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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 20449 = 11^{2} \cdot 13^{2} \) |
| $3$ | split multiplicative | $4$ | \( 20449 = 11^{2} \cdot 13^{2} \) |
| $11$ | additive | $62$ | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
| $13$ | additive | $74$ | \( 726 = 2 \cdot 3 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 122694de
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 1014f2, its twist by $-143$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.676.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.1827904.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.2244729488859.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.2.16422346512.3 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.138986698372081800194468370379488752996352.2 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.46328899457360600064822790126496250998784.1 | \(\Z/6\Z\) | 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 | split | split | ord | ord | add | add | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 9 | 3 | 0 | 0 | - | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 | - | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.