Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-9101x+385923\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-9101xz^2+385923z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-11794923x+18182555622\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-41, 852)$ | $0.14367985671545179594438009237$ | $\infty$ |
Integral points
\( \left(-41, 852\right) \), \( \left(-41, -812\right) \), \( \left(7, 564\right) \), \( \left(7, -572\right) \), \( \left(63, 228\right) \), \( \left(63, -292\right) \), \( \left(87, 468\right) \), \( \left(87, -556\right) \)
Invariants
| Conductor: | $N$ | = | \( 6370 \) | = | $2 \cdot 5 \cdot 7^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-17366515712000$ | = | $-1 \cdot 2^{24} \cdot 5^{3} \cdot 7^{2} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( -\frac{1701366814932001}{354418688000} \) | = | $-1 \cdot 2^{-24} \cdot 5^{-3} \cdot 7 \cdot 13^{-2} \cdot 17^{3} \cdot 3671^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2623066307650057678430851446$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.93798827258912021699219302069$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9654864870087926$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.48314858939427$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.14367985671545179594438009237$ |
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| Real period: | $\Omega$ | ≈ | $0.66272759938277779105132631429$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ ( 2^{3} \cdot 3 )\cdot1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.5705891129932569487312960476 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.570589113 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.662728 \cdot 0.143680 \cdot 48}{1^2} \\ & \approx 4.570589113\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 12672 |
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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 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 |
| $5$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $7$ | $1$ | $II$ | additive | -1 | 2 | 2 | 0 |
| $13$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 415 & 6 \\ 414 & 7 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 337 & 6 \\ 171 & 19 \end{array}\right),\left(\begin{array}{rr} 211 & 6 \\ 213 & 19 \end{array}\right),\left(\begin{array}{rr} 246 & 181 \\ 385 & 351 \end{array}\right),\left(\begin{array}{rr} 303 & 2 \\ 310 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[420])$ is a degree-$278691840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/420\Z)$.
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 | $4$ | \( 245 = 5 \cdot 7^{2} \) |
| $3$ | good | $2$ | \( 637 = 7^{2} \cdot 13 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 1274 = 2 \cdot 7^{2} \cdot 13 \) |
| $7$ | additive | $14$ | \( 130 = 2 \cdot 5 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 490 = 2 \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 6370n
consists of 2 curves linked by isogenies of
degree 3.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.980.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.19208000.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.12960667629.4 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.6722800.1 | \(\Z/6\Z\) | 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 |
| $18$ | 18.0.2823221462879788350940287000000000000.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.2.139335600731863049437603167948096000000.2 | \(\Z/6\Z\) | 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 | ord | nonsplit | add | ss | nonsplit | ss | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 1 | 1 | - | 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 | 0 | 0 | 0 |
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.