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Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2+1236x-72604\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z+1236xz^2-72604z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+19773x-4626882\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 3042 \) | = | $2 \cdot 3^{2} \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $-2378670782436$ | = | $-1 \cdot 2^{2} \cdot 3^{6} \cdot 13^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{351}{4} \) | = | $2^{-2} \cdot 3^{3} \cdot 13$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0557529082976339847370798804$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.2035194743441120183295343658$ |
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| $abc$ quality: | $Q$ | ≈ | $1.2727904629543532$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.476179079335514$ | |||
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: | $\Omega$ | ≈ | $0.40191124794702138842433455917$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.6076449917880855536973382367 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.607644992 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.401911 \cdot 1.000000 \cdot 4}{1^2} \\ & \approx 1.607644992\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4992 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $2$ | $I_0^{*}$ | 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 | 4.8.0.2 |
| $7$ | 7B | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1092 = 2^{2} \cdot 3 \cdot 7 \cdot 13 \), index $768$, genus $21$, and generators
$\left(\begin{array}{rr} 1073 & 336 \\ 84 & 305 \end{array}\right),\left(\begin{array}{rr} 421 & 441 \\ 693 & 106 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 364 & 1 \end{array}\right),\left(\begin{array}{rr} 169 & 168 \\ 924 & 169 \end{array}\right),\left(\begin{array}{rr} 379 & 735 \\ 336 & 169 \end{array}\right),\left(\begin{array}{rr} 1 & 936 \\ 336 & 1 \end{array}\right),\left(\begin{array}{rr} 727 & 0 \\ 0 & 1091 \end{array}\right),\left(\begin{array}{rr} 1 & 756 \\ 0 & 157 \end{array}\right),\left(\begin{array}{rr} 925 & 0 \\ 0 & 85 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 924 & 1 \end{array}\right),\left(\begin{array}{rr} 729 & 364 \\ 728 & 729 \end{array}\right)$.
The torsion field $K:=\Q(E[1092])$ is a degree-$317011968$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1092\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 1521 = 3^{2} \cdot 13^{2} \) |
| $3$ | additive | $6$ | \( 338 = 2 \cdot 13^{2} \) |
| $13$ | additive | $74$ | \( 18 = 2 \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 3042b
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 338a1, its twist by $-39$.
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 |
|---|---|---|---|
| $3$ | 3.1.676.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.1827904.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.49353408.1 | \(\Z/4\Z\) | not in database |
| $6$ | 6.0.12338352.1 | \(\Z/4\Z\) | not in database |
| $6$ | 6.0.771147.1 | \(\Z/7\Z\) | not in database |
| $8$ | 8.2.999406512.2 | \(\Z/3\Z\) | not in database |
| $12$ | 12.0.2435758881214464.2 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $18$ | 18.0.1878328153971890270208.2 | \(\Z/28\Z\) | not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | add | ord | ord | ord | add | ord | ss | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 8 | - | 2 | 0 | 0 | - | 0 | 0,0 | 0 | 2 | 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 $0$.