Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2+345483x-350275549\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z+345483xz^2-350275549z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+27984096x-255434827536\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 8624 \) | = | $2^{4} \cdot 7^{2} \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $-55677334351972315136$ | = | $-1 \cdot 2^{12} \cdot 7^{8} \cdot 11^{9} $ |
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| j-invariant: | $j$ | = | \( \frac{9463555063808}{115539436859} \) | = | $2^{15} \cdot 7^{-2} \cdot 11^{-9} \cdot 661^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4697148743703075168900726769$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.80361261928270555492016418372$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0659265883154396$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.834306886538049$ | |||
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: | $\Omega$ | ≈ | $0.097572162782727689350148337067$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 18 $ = $ 1\cdot2\cdot3^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.7562989300890984083026700672 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.756298930 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.097572 \cdot 1.000000 \cdot 18}{1^2} \\ & \approx 1.756298930\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 207360 |
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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 |
| $7$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $11$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
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 |
|---|---|---|
| $3$ | 3B | 9.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2772 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 11 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2771 & 2754 \\ 0 & 2617 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 1403 & 2754 \\ 1404 & 2753 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 2765 & 2754 \\ 1602 & 2309 \end{array}\right),\left(\begin{array}{rr} 2510 & 2763 \\ 1899 & 1394 \end{array}\right),\left(\begin{array}{rr} 1385 & 0 \\ 0 & 2771 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 2755 & 18 \\ 2754 & 19 \end{array}\right)$.
The torsion field $K:=\Q(E[2772])$ is a degree-$68976230400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2772\Z)$.
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 | $2$ | \( 539 = 7^{2} \cdot 11 \) |
| $3$ | good | $2$ | \( 784 = 2^{4} \cdot 7^{2} \) |
| $7$ | additive | $32$ | \( 176 = 2^{4} \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 784 = 2^{4} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 8624bb
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 77b2, its twist by $28$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-21}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.44.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.21296.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.784147392.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.21171979584.1 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.286868736.2 | \(\Z/6\Z\) | 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 |
| $12$ | 12.0.9148014683781894144.3 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.2.54667497285142944140492468599652352.1 | \(\Z/6\Z\) | not in database |
| $18$ | 18.0.1076020349063468569517313259446957244416.1 | \(\Z/18\Z\) | 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 | ord | ord | add | split | ord | ord | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 2 | 0 | - | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 |
| $\mu$-invariant(s) | - | 2 | 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 $0$.