Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-108924x-4622883\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-108924xz^2-4622883z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-141165531x-213567737994\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-51, 921)$ | $1.4826204323138279040183628211$ | $\infty$ |
| $(-307, 153)$ | $0$ | $2$ |
Integral points
\( \left(-307, 153\right) \), \( \left(-51, 921\right) \), \( \left(-51, -871\right) \), \( \left(369, 2181\right) \), \( \left(369, -2551\right) \), \( \left(3293, 186393\right) \), \( \left(3293, -189687\right) \)
Invariants
| Conductor: | $N$ | = | \( 78078 \) | = | $2 \cdot 3 \cdot 7 \cdot 11 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $73656750704689152$ | = | $2^{20} \cdot 3^{3} \cdot 7^{2} \cdot 11 \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{29609739866953}{15259926528} \) | = | $2^{-20} \cdot 3^{-3} \cdot 7^{-2} \cdot 11^{-1} \cdot 30937^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9286326532720435915033557376$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.64615797454127522347661201682$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0045488905924413$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.119565979754695$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.4826204323138279040183628211$ |
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| Real period: | $\Omega$ | ≈ | $0.27786597666600996295489974091$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 80 $ = $ ( 2^{2} \cdot 5 )\cdot1\cdot2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.2393954889972741609521333664 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.239395489 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.277866 \cdot 1.482620 \cdot 80}{2^2} \\ & \approx 8.239395489\end{aligned}$$
Modular invariants
Modular form 78078.2.a.bz
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1105920 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $20$ | $I_{20}$ | split multiplicative | -1 | 1 | 20 | 20 |
| $3$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 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 |
|---|---|---|
| $2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 976 & 2379 \\ 2821 & 794 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 3426 & 3427 \end{array}\right),\left(\begin{array}{rr} 791 & 0 \\ 0 & 3431 \end{array}\right),\left(\begin{array}{rr} 2692 & 793 \\ 2015 & 534 \end{array}\right),\left(\begin{array}{rr} 3425 & 8 \\ 3424 & 9 \end{array}\right),\left(\begin{array}{rr} 833 & 1092 \\ 2678 & 1223 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 1691 & 1950 \\ 962 & 2939 \end{array}\right)$.
The torsion field $K:=\Q(E[3432])$ is a degree-$531372441600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3432\Z)$.
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 | $4$ | \( 5577 = 3 \cdot 11 \cdot 13^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 26026 = 2 \cdot 7 \cdot 11 \cdot 13^{2} \) |
| $5$ | good | $2$ | \( 39039 = 3 \cdot 7 \cdot 11 \cdot 13^{2} \) |
| $7$ | split multiplicative | $8$ | \( 11154 = 2 \cdot 3 \cdot 11 \cdot 13^{2} \) |
| $11$ | nonsplit multiplicative | $12$ | \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \) |
| $13$ | additive | $86$ | \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 78078ci
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 462b1, its twist by $13$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-39}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-143}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{33}, \sqrt{-39})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.10319205371904.9 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | nonsplit | ord | split | nonsplit | add | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | 1 | 1 | 2 | 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 |
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.