Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-136261488x+612220410736\)
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(homogenize, simplify) |
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\(y^2z=x^3-136261488xz^2+612220410736z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-136261488x+612220410736\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 17424 \) | = | $2^{4} \cdot 3^{2} \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $-58188380811264$ | = | $-1 \cdot 2^{12} \cdot 3^{6} \cdot 11^{7} $ |
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| j-invariant: | $j$ | = | \( -\frac{52893159101157376}{11} \) | = | $-1 \cdot 2^{12} \cdot 11^{-1} \cdot 29^{3} \cdot 809^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9381100763918325497571919341$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.49670911509864712261136540520$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0929566831983986$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.943130503587764$ | |||
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: | $\Omega$ | ≈ | $0.25394737714587441585583798371$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.50789475429174883171167596741 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.507894754 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.253947 \cdot 1.000000 \cdot 2}{1^2} \\ & \approx 0.507894754\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 720000 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II^{*}$ | additive | -1 | 4 | 12 | 0 |
| $3$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 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.4.2 | 25.60.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 \), index $1200$, genus $37$, and generators
$\left(\begin{array}{rr} 599 & 1050 \\ 600 & 1049 \end{array}\right),\left(\begin{array}{rr} 764 & 2055 \\ 585 & 884 \end{array}\right),\left(\begin{array}{rr} 38 & 41 \\ 741 & 539 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 50 & 1 \end{array}\right),\left(\begin{array}{rr} 1099 & 0 \\ 0 & 3299 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1540 & 9 \\ 1227 & 2413 \end{array}\right),\left(\begin{array}{rr} 2162 & 1059 \\ 909 & 1661 \end{array}\right),\left(\begin{array}{rr} 1649 & 0 \\ 0 & 3299 \end{array}\right),\left(\begin{array}{rr} 3251 & 50 \\ 3250 & 51 \end{array}\right)$.
The torsion field $K:=\Q(E[3300])$ is a degree-$15206400000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3300\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 1089 = 3^{2} \cdot 11^{2} \) |
| $3$ | additive | $6$ | \( 1936 = 2^{4} \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 144 = 2^{4} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5 and 25.
Its isogeny class 17424bv
consists of 3 curves linked by isogenies of
degrees dividing 25.
Twists
The minimal quadratic twist of this elliptic curve is 11a2, its twist by $-132$.
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 |
| $4$ | 4.4.2178000.1 | \(\Z/5\Z\) | not in database |
| $6$ | 6.0.21296.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.991847400192.2 | \(\Z/3\Z\) | not in database |
| $10$ | 10.0.5729812889130000000000.1 | \(\Z/5\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/10\Z\) | not in database |
| $20$ | 20.20.1001915153333748708256741333007812500000000000000000000.4 | \(\Z/25\Z\) | 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 | add | add | ord | ord | add | ord | ord | ss | ord | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 0 | 0 | - | 0 | 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 | 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 $0$.